#208159

order	512
nilpotency class	3
isoclinism	isoclinic class of #162118
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ch,d^b=dh,e^b=ei,g^b=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, 115355);