C₂×#8705 #189699

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#8705

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1322 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 1322 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 256

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96895);