#6368749

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	#62487	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)(33,34,38,42)(35,39,44,53)(36,46,48,56)(37,49,51,41)(40,50,55,47)(43,57,58,64)(45,61,60,54)(52,62,63,59);(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)(34,39)(36,43)(37,45)(38,44)(40,52)(41,54)(42,53)(46,57)(47,59)(48,58)(49,61)(50,62)(51,60)(55,63)(56,64);(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)(33,36,38,48)(34,40,42,55)(35,43,44,58)(37,50,51,47)(39,52,53,63)(41,56,49,46)(45,62,60,59)(54,64,61,57);(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)(34,39)(36,43)(37,45)(38,44)(40,52)(41,54)(42,53)(46,57)(47,59)(48,58)(49,61)(50,62)(51,60)(55,63)(56,64);(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)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,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	#72306685, #72306699, #72306707
maximal normal subgroup of	#72306685, #72306699, #72306707

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275945);