#409388

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1304 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)(7,9,11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,30)(24,29)(25,33)(26,28)(27,31)(32,36)(34,35);(5,7)(6,9)(8,11)(10,12)(13,15,16,19)(14,17,18,20)(21,23)(22,27)(24,31)(25,34)(26,32)(28,35)(29,36)(30,33);(1,2,3,4)(7,11)(9,12)(22,28)(27,35)(29,33)(30,36);(14,18)(17,20)(21,24,26,25)(22,29,28,33)(23,31,32,34)(27,36,35,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei

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, 316584);