#7534565

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#45057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1679 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7441761);