C₂×#13378 #498987

order	512
nilpotency class	4
isoclinism	isoclinic class of #13357
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13378

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1698 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1698 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 26
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4,6,8)(5,9,10,7)(12,16)(13,19)(14,22)(17,25)(18,20)(23,26);(3,5,6,10)(4,7,8,9)(12,17)(14,22)(16,25)(18,23)(20,26)(21,24);(11,12)(13,20)(14,23)(15,18)(16,19)(17,22)(21,25)(24,26)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#13378⋊C₃
maximal subgroup of	C₂×#11714280, #72213149, #72213520, …
maximal normal subgroup of	#72213149, #72213520, #72261949, …
maximal quotient of	C₂×#144629, C₂²×#13378, C₂×#149485, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406183);