C₂×#31076 #7503369

order	512
nilpotency class	3
isoclinism	isoclinic class of #31076
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#31076

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	#2679
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3542 conjugacy classes of subgroups; only the 586 normal subgroups are shown.

normal subgroups only

This group has 3542 conjugacy classes of subgroups — too many to draw. Only its 586 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) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 18 points:

(1,2);(4,10,12,11)(5,13,15,18);(3,4)(5,14)(6,13)(7,11)(8,10)(9,12)(15,16)(17,18);(3,5)(4,6)(7,13)(8,18)(9,15)(10,14)(11,16)(12,17);(3,6)(4,5)(7,16)(8,14)(9,17)(10,18)(11,13)(12,15)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#31076⋊C₃
maximal subgroup of	C₂²×#31076, #72258208, #72258215, …
maximal normal subgroup of	C₂²×#31076, #72258208, #72258215, …
maximal quotient of	C₂²×#31076, #72258215, #72258217, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410565);