#7620321

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24780	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 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)(3,7)(4,8)(5,6)(10,14)(11,12)(17,18,23,27)(19,28,30,24)(20,31,26,22)(21,32,29,25);(2,6)(7,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,24)(20,32)(21,31)(22,29)(23,30)(25,26)(27,28);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,23,26)(18,22,27,31)(19,25,30,32)(21,28,29,24);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(17,21,31,19,23,29,22,30)(18,25,26,24,27,32,20,28);(2,6)(7,8)(9,12,13,11)(10,15,14,16)(17,22,23,31)(18,26,27,20)(19,29,30,21)(24,25,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527517);