C₂²×#2862 #7633337

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂²×#2862

Subgroups

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

10434 conjugacy classes of subgroups; only the 4770 normal subgroups are shown.

normal subgroups only

This group has 10434 conjugacy classes of subgroups — too many to draw. Only its 4770 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) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#36415, C₂²×#36149, C₂²×#39618, …
maximal normal subgroup of	C₂²×#36415, C₂²×#36149, C₂²×#39618, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540533);