#417098

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,c^2=f,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^c=di

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, 324294);