#915108

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2046 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=ghi,d^2=g,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 822304);