C₂²×#2844 #7633390

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂²×#2844

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

6554 conjugacy classes of subgroups; only the 3490 normal subgroups are shown.

normal subgroups only

This group has 6554 conjugacy classes of subgroups — too many to draw. Only its 3490 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) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#47818, C₂²×#47704
maximal normal subgroup of	C₂²×#47818, C₂²×#47704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540586);