#6368765

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368694
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62551	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1449 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2,8,13)(3,14,18,30)(4,5,10,23)(6,12,7,11)(9,25,29,17)(15,16,19,32)(20,26,24,22)(21,31,27,28)(33,34)(35,41)(36,47)(37,42)(38,51)(39,45)(40,49)(43,50)(44,59)(46,54)(48,58)(52,56)(53,63)(55,61)(57,62)(60,64);(1,3,8,18)(2,9,13,29)(4,19,10,15)(5,21,23,27)(6,17,7,25)(11,14,12,30)(16,26,32,22)(20,31,24,28)(33,35,37,45)(34,39,42,41)(36,44,48,60)(38,46,50,62)(40,53,56,55)(43,54,51,57)(47,64,58,59)(49,61,52,63);(1,4,8,10)(2,5,13,23)(3,15,18,19)(6,24,7,20)(9,27,29,21)(11,26,12,22)(14,16,30,32)(17,31,25,28)(33,36,37,48)(34,40,42,56)(35,44,45,60)(38,52,50,49)(39,53,41,55)(43,58,51,47)(46,63,62,61)(54,59,57,64);(1,5)(2,10)(3,16)(4,13)(6,22)(7,26)(8,23)(9,28)(11,24)(12,20)(14,19)(15,30)(17,27)(18,32)(21,25)(29,31)(33,35)(34,41)(36,44)(37,45)(38,46)(39,42)(40,55)(43,57)(47,59)(48,60)(49,61)(50,62)(51,54)(52,63)(53,56)(58,64);(1,6,8,7)(2,11,13,12)(3,17,18,25)(4,20,10,24)(5,22,23,26)(9,14,29,30)(15,28,19,31)(16,27,32,21)(35,45)(39,41)(44,60)(46,62)(53,55)(54,57)(59,64)(61,63)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306666, #72306694, #72306707
maximal normal subgroup of	#72306666, #72306694, #72306707

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275961);