#158992

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

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, 66188);