#189694

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dh,g^c=gh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96890);