#206953

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1026 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)(3,7)(4,5)(6,8)(10,14,15,16)(17,18)(19,25)(20,29)(21,23)(22,27)(24,26)(28,31)(30,32);(2,5)(6,7)(9,10,12,16,13,15,11,14)(17,19,26,32,21,28,20,27)(18,22,29,31,23,30,24,25);(1,3,4,8)(2,6,5,7)(18,23)(22,30)(24,29)(25,31);(9,11,13,12)(10,14,15,16)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fh,d^2=i,e^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi

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, 114149);