C₂×#12774 #493606

order	512
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#12774

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×H(3,𝔽₄)	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

3136 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 3136 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) = 22
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#11714276, #72271414, #72271422
maximal normal subgroup of	#72271414, #72271422
maximal quotient of	C₂×#149485

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400802);