#876976

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2292 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)(3,8)(4,7)(5,6)(10,13)(11,15)(17,18,21,24)(19,23,20,22)(26,29)(27,31);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(26,29)(27,31);(1,2)(3,7)(4,8)(5,6)(25,26)(27,30)(28,29)(31,32);(1,3,5,4)(2,7,6,8)(17,19,21,20)(18,22,24,23)(25,27,28,31)(26,30,29,32);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,23,24,22)

Polycyclic presentation:

2^9|a^2=hi,d^2=i,e^2=f,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, 784172);