#1271374

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1702 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=fg,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1178570);