#226032

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
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	C₂²×D₈	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

688 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 133228);