C₂×#12363 #489056

order	512
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12363

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

705 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 705 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 4 generators on 42 points:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396252);