C₂×#12971 #494382

order	512
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12971

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1112 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 258 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);(3,4,6,5)(7,8)(9,19)(10,16)(11,24)(12,27)(13,18)(14,20)(15,23)(17,29)(21,37)(22,34)(25,32)(26,33)(28,30)(31,38)(35,36);(3,5,6,4)(7,9)(8,14)(10,20)(11,25)(12,28)(13,22)(15,30)(16,34)(17,31)(18,32)(19,24)(21,29)(23,38)(26,36)(27,35)(33,37);(7,10,13,11)(8,15,18,33)(9,20,22,25)(12,26,29,23)(14,30,32,37)(16,17,24,27)(19,35,34,31)(21,38,28,36)

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, 401578);