#7610788

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#23962	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,23)(21,25)(22,24)(27,31)(28,29)(30,32);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,25)(18,21,24,26)(20,23,30,32)(27,31,28,29);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19)(18,21)(20,28)(22,25)(23,31)(24,26)(27,30)(29,32);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,20,21,29,22,30,26,31)(18,23,25,28,24,32,19,27);(2,8)(6,7)(17,21,22,26)(18,19,24,25)(20,29,30,31)(23,27,32,28)

Polycyclic presentation:

2^9|b^2=f,c^2=fi,d^2=fhi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=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, 7517984);