C₂×#12239 #484440

order	512
nilpotency class	4
isoclinism	isoclinic class of #12099
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#12239

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

969 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 969 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 34
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391636);