#158989

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×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)(9,10,13,11)(12,14,16,15)(17,18)(19,22)(20,21)(23,24)(25,26,29,32)(27,31,28,30);(2,4)(6,8)(10,11)(12,16)(18,20)(22,24)(25,27,29,28)(26,30,32,31);(9,11,13,10)(12,14,16,15)(17,19,21,23)(18,22,20,24)(25,28,29,27)(26,31,32,30);(1,2)(3,4)(5,6)(7,8)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,22,23,24)(25,26)(27,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,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, 66185);