#391893

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂³×C₈
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1274 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 1274 conjugacy classes of subgroups — too many to draw. Only its 370 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=e,d^2=e,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299089);