#6357454

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357427
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1865 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,7,11)(3,12,13,4)(5,10,6,16)(8,9,15,14)(17,18,22,26)(19,27,30,41)(20,24,32,38)(21,34,28,25)(23,35,37,31)(29,40,43,48)(33,45,42,39)(36,46,47,44);(1,3,5,9,7,13,6,14)(2,8,16,4,11,15,10,12)(17,19,28,35,22,30,21,31)(18,23,34,41,26,37,25,27)(20,29,42,46,32,43,33,44)(24,36,45,48,38,47,39,40);(1,4,6,8,7,12,5,15)(2,9,10,3,11,14,16,13)(17,20)(18,24)(19,29)(21,33)(22,32)(23,36)(25,39)(26,38)(27,40)(28,42)(30,43)(31,44)(34,45)(35,46)(37,47)(41,48);(3,13)(4,12)(8,15)(9,14)(20,32)(24,38)(29,43)(33,42)(36,47)(39,45)(40,48)(44,46);(1,5,7,6)(2,10,11,16)(3,14,13,9)(4,15,12,8)(19,30)(23,37)(27,41)(29,43)(31,35)(36,47)(40,48)(44,46)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,e^2=i,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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306858, #72306859, #72306928
maximal normal subgroup of	#72306858, #72306859, #72306928

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264650);