#6368914

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368846
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1515 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306703, #72306708, #72306718
maximal normal subgroup of	#72306703, #72306708, #72306718

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276110);