#204733

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1064 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111929);