#6407914

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2819	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1342 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,21,25,22,24,30,31)(19,23,27,20,26,28,32,29);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,28)(29,31)(30,32);(2,6)(3,7)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,28)(29,31)(30,32);(2,6)(5,8)(10,13)(14,16)(18,24)(19,26)(25,31)(27,32);(17,20,21,28,22,29,30,23)(18,19,25,27,24,26,31,32)

Polycyclic presentation:

2^9|a^2=h,c^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=cfg,d^b=dfi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6315110);