#6368764

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	#62550	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

(2,8)(5,6)(9,22)(10,23)(11,20)(12,25)(14,19)(16,26)(17,21)(24,28)(27,32)(29,30)(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);(1,2,7,11)(3,12,15,9)(4,16,18,23)(5,8,6,20)(10,21,26,17)(13,28,31,32)(14,22,19,25)(24,30,27,29)(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);(1,3,7,15)(2,9,11,12)(4,13,18,31)(5,19,6,14)(8,22,20,25)(10,24,26,27)(16,32,23,28)(17,30,21,29)(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);(1,4)(2,10)(3,13)(5,17)(6,21)(7,18)(8,23)(9,24)(11,26)(12,27)(14,29)(15,31)(16,20)(19,30)(22,28)(25,32)(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);(1,5,7,6)(2,8,11,20)(3,14,15,19)(4,17,18,21)(9,25,12,22)(10,23,26,16)(13,29,31,30)(24,32,27,28)(35,45)(39,41)(44,60)(46,62)(53,55)(54,57)(59,64)(61,63)

Polycyclic presentation:

2^9|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	#72306547, #72306666, #72306708
maximal normal subgroup of	#72306547, #72306666, #72306708

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275960);