#204152

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
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×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1123 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)(6,10)(8,16)(11,17)(12,18)(13,20)(15,19)(21,22,25,28,26,29,24,34)(23,30,33,35,31,36,32,27);(5,6,8,10,9,12,16,18)(7,13,15,20,14,11,19,17)(21,23)(22,27,29,35)(24,32)(25,33)(26,31)(28,30,34,36);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,17)(12,13)(16,19)(18,20)(23,31)(27,35)(30,36)(32,33);(7,14)(11,13)(15,19)(17,20)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ch,d^c=dhi

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, 111348);