C₂×#12775 #493607

order	512
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#12775

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2384 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 2384 conjugacy classes of subgroups — too many to draw. Only its 322 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)(7,8,12,15,13,18,11,17)(9,19,22,36,23,34,21,37)(10,24,27,33,28,16,26,35)(14,29,30,25,32,38,31,20);(7,9,13,23)(8,14,18,32)(10,25,28,20)(11,22,12,21)(15,30,17,31)(16,34,24,19)(26,38,27,29)(33,36,35,37);(4,6)(8,15)(11,12)(14,30)(16,35)(17,18)(19,36)(21,22)(24,33)(26,27)(29,38)(31,32)(34,37)

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, 400803);