#415537

order	512
nilpotency class	3
isoclinism	isoclinic class of #394183
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1116 conjugacy classes of subgroups; only the 330 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 330 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=gh,b^2=e,c^2=fi,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=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, 322733);