#207943

order	512
nilpotency class	3
isoclinism	isoclinic class of #161854
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

891 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)(10,14)(11,12)(15,16)(17,18)(19,22,26,31)(20,27)(21,24)(23,29)(25,32,28,30);(9,10,12,14,13,15,11,16)(17,19)(18,22,24,31)(20,28)(21,26)(23,32,27,30)(25,29);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32);(1,3,4,7)(2,6,5,8)(20,29)(23,27)(25,28)(30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=fh,e^2=hi,b^a=be,c^a=cf,e^a=eh,c^b=ch,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115139);