#204146

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1156 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)(7,14)(8,13)(9,18)(10,12)(11,15)(16,20)(17,19)(21,22,26,30,25,31,24,29)(23,32,33,36,35,28,34,27);(5,7,9,15,10,16,8,17)(6,11,13,14,12,19,18,20)(21,23)(22,27,31,36)(24,34)(25,35)(26,33)(28,30,32,29);(1,2,3,4)(6,12)(11,19)(13,18)(14,20)(21,23)(22,28)(24,34)(25,35)(26,33)(27,30)(29,36)(31,32);(5,8,10,9)(6,13,12,18)(7,15,16,17)(11,20,19,14)(21,24,25,26)(22,29,31,30)(23,33,35,34)(27,32,36,28)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh,d^c=di

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, 111342);