C₂×#11807 #481830

order	512
nilpotency class	4
isoclinism	isoclinic class of #11799
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#11807

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1046 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 210 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)(5,7)(6,8)(9,10)(11,12,15,20,16,19,28,35)(13,21,23,36,24,17,29,31)(14,18,26,34,27,33,39,42)(22,32,37,41,25,30,38,40);(4,7)(5,10)(8,9)(11,13,16,24)(12,17,19,21)(14,25,27,22)(15,23,28,29)(18,32,33,30)(20,36,35,31)(26,38,39,37)(34,40,42,41);(4,8)(7,9)(11,14,16,27)(12,18,19,33)(13,22,24,25)(15,26,28,39)(17,30,21,32)(20,34,35,42)(23,37,29,38)(31,40,36,41)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389026);