C₂×#12704 #493381

order	512
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#12704

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

3354 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 3354 conjugacy classes of subgroups — too many to draw. Only its 338 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) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(3,4)(5,6)(7,8,12,17,13,18,10,15)(9,19,22,37,23,34,20,36)(11,25,24,35,27,16,28,33)(14,29,30,26,31,38,32,21);(7,9,13,23)(8,14,18,31)(10,22,12,20)(11,26,27,21)(15,32,17,30)(16,34,25,19)(24,29,28,38)(33,36,35,37);(4,6)(7,10,13,12)(8,15,18,17)(9,20,23,22)(11,24,27,28)(14,30,31,32)(16,33,25,35)(19,36,34,37)(21,29,26,38)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400577);