#7607419

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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
prime radical	#28916	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=ghi,d^2=i,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7514615);