#231588

order	512
nilpotency class	3
isoclinism	isoclinic class of #19883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,10,11,9,12,8,14)(7,15,18,20,17,13,16,19)(21,22)(23,30)(24,28)(25,34)(26,29)(27,32)(31,35)(33,36);(1,3,4,2)(6,11)(7,16)(8,10)(12,14)(17,18)(19,20);(6,12)(11,14)(13,15)(19,20)(21,23,25,32,26,33,24,31)(22,27,28,30,29,35,34,36);(5,7,8,16,9,17,10,18)(6,13,14,20,12,15,11,19)(21,24,26,25)(22,28,29,34)(23,31,33,32)(27,30,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,d^b=de,e^b=eh,g^b=gh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138784);