C₂²×#2645 #7627775

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#2645

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

6322 conjugacy classes of subgroups; only the 3490 normal subgroups are shown.

normal subgroups only

This group has 6322 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);(5,6,9,7)(8,10,12,11)(15,19)(17,20)(21,22,25,23)(24,27,28,26);(6,7)(10,11)(13,14,16,18)(15,20,19,17)(21,23,25,22)(24,27,28,26);(5,7,9,6)(8,10,12,11)(13,15)(14,17)(16,19)(18,20)(21,24)(22,26)(23,27)(25,28);(5,8,9,12)(6,10,7,11)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#53098, C₂²×#39146, C₂²×#42598, …
maximal normal subgroup of	C₂²×#53098, C₂²×#39146, C₂²×#42598, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534971);