#223113

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

869 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=eh,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130309);