#7080736

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,24,22,26,21,20)(19,27,30,32,28,25,31,29);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,28)(20,26)(21,23)(25,32)(27,29);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,19,22,28)(18,25,26,27)(20,32,24,29)(21,31,23,30);(3,8)(5,7)(11,16)(13,15)(17,20,23,18,22,24,21,26)(19,29,30,27,28,32,31,25);(10,14)(13,15)(17,21,22,23)(18,24,26,20)(19,30,28,31)(25,32,27,29)

Polycyclic presentation:

2^9|a^2=fg,c^2=h,d^2=fg,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308949, #72308965
maximal normal subgroup of	#72308949, #72308965

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6987932);