C₂²×#2823 #7633397

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#2823

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

9138 conjugacy classes of subgroups; only the 3490 normal subgroups are shown.

normal subgroups only

This group has 9138 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) = 24
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#47678, C₂²×#54025, C₂²×#53653, …
maximal normal subgroup of	C₂²×#47678, C₂²×#54025, C₂²×#53653, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540593);