C₂×#32899 #6628889

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#32899

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

17872 conjugacy classes of subgroups; only the 954 normal subgroups are shown.

normal subgroups only

This group has 17872 conjugacy classes of subgroups — too many to draw. Only its 954 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 — 5 generators on 22 points:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#32899⋊C₃, C₂×#72245678, C₂×#72245714, …
maximal subgroup of	C₂×#6470228, #72235210, #72235216, …
maximal normal subgroup of	C₂×#6470228, #72235210, #72235216, …
maximal quotient of	#72235768, #72237396, C₂×#511853, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536085);