#208448

order	512
nilpotency class	3
isoclinism	isoclinic class of #161570
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1098 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=f,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115644);