#216199

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 123395);