#375547

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1314 conjugacy classes of subgroups; only the 402 normal subgroups are shown.

normal subgroups only

This group has 1314 conjugacy classes of subgroups — too many to draw. Only its 402 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)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22,26,29,25,28,34,24)(23,27,32,30,31,35,36,33);(5,7,9,8)(6,10,12,11)(13,15,16,19)(14,17,18,20)(21,23,25,31)(22,27,28,35)(24,33,29,30)(26,32,34,36);(1,2,3,4)(5,7,9,8)(6,11,12,10)(14,18)(15,19)(21,24,34,28,25,29,26,22)(23,30,36,27,31,33,32,35);(5,8,9,7)(6,11,12,10)(23,31)(27,35)(30,33)(32,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=efgh,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282743);