C₂×#12862 #493873

order	512
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12862

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁴×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1718 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1718 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) = 42
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 42 points:

(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,15,20,16,21,28,35)(13,17,24,30,22,31,37,42)(14,19,26,34,25,18,27,33)(23,29,38,40,36,32,39,41);(4,7)(8,10)(11,13)(12,17)(14,23)(15,24)(16,22)(18,32)(19,29)(20,30)(21,31)(25,36)(26,38)(27,39)(28,37)(33,41)(34,40)(35,42);(3,5,6,9)(4,8,7,10)(12,18)(13,22)(14,25)(17,29)(19,21)(20,34)(24,37)(26,27)(30,41)(31,32)(33,35)(40,42)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266756, #72266760, #72266868, …
maximal normal subgroup of	#72266756, #72266760, #72266868, …
maximal quotient of	C₂×#148577, #72267438, #72267459

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401069);