#188847

order	512
nilpotency class	3
isoclinism	isoclinic class of #188818
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,g^c=gi

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, 96043);