C₂×#12099 #484172

order	512
nilpotency class	4
isoclinism	isoclinic class of #12099
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12099

Subgroups

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

1467 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1467 conjugacy classes of subgroups — too many to draw. Only its 146 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)(5,7)(6,9)(8,10)(11,12)(13,16)(14,17)(15,19)(18,23)(20,25)(21,26)(22,24);(4,7)(5,10)(8,9)(11,13)(12,16)(14,22)(15,20)(17,26)(18,24)(19,25)(21,23);(12,17)(13,20)(14,23)(16,24)(18,19)(25,26)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282723
maximal normal subgroup of	#72282723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391368);