#6354823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354774
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1486 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306999, #72307005, #72307064
maximal normal subgroup of	#72306999, #72307005, #72307064

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262019);