C₂×#12218 #484316

order	512
nilpotency class	4
isoclinism	isoclinic class of #12093
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12218

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁶	D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1267 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1267 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) = 34
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227782
maximal normal subgroup of	#72227782
maximal quotient of	#72227776, #72227782

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391512);