C₂×#28261 #6465477

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#28261

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2606 conjugacy classes of subgroups; only the 698 normal subgroups are shown.

normal subgroups only

This group has 2606 conjugacy classes of subgroups — too many to draw. Only its 698 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) = 512

A minimal faithful permutation representation — 5 generators on 42 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372673);