C₂×#11509 #480260

order	512
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#11509

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂⁴×D₄	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

2194 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 2194 conjugacy classes of subgroups — too many to draw. Only its 234 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);(3,4,5,8,6,7,9,10)(11,12)(13,17)(14,18,22,16)(15,19)(20,25)(21,24,26,23);(12,16)(13,20)(14,22)(17,24)(18,19)(23,25);(4,7)(8,10)(11,13,15,20)(12,17,19,25)(14,21,22,26)(16,23,18,24)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#134049, #72284365, #72284436, …
maximal normal subgroup of	C₂×#134049, #72284365, #72284436, …
maximal quotient of	#72284365

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387456);