C₂×#23565 #6431999

order	512
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#23565

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4374 conjugacy classes of subgroups; only the 994 normal subgroups are shown.

normal subgroups only

This group has 4374 conjugacy classes of subgroups — too many to draw. Only its 994 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 — 5 generators on 26 points:

(1,2);(3,4,7,9)(5,10,6,8)(11,12)(13,16)(14,17)(15,18)(19,24)(20,23)(21,25)(22,26);(3,5,7,6)(4,8,9,10)(11,13)(12,16)(14,19)(15,20)(17,24)(18,23)(21,26)(22,25);(3,6,7,5)(4,8,9,10)(11,14)(12,17)(13,19)(15,21)(16,22)(18,25)(20,26)(23,24);(14,21)(17,25)(19,26)(22,24)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267778
maximal normal subgroup of	#72267778
maximal quotient of	#72267781

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339195);