C₂²×#2860 #7633334

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#2860

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

10314 conjugacy classes of subgroups; only the 3906 normal subgroups are shown.

normal subgroups only

This group has 10314 conjugacy classes of subgroups — too many to draw. Only its 3906 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);(6,8)(10,11)(14,15)(19,22)(20,24);(9,10,13,11)(12,14,16,15)(17,18,21,23)(19,20,22,24);(5,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,21,22)(18,20,23,24);(9,12,13,16)(10,14,11,15)(17,20)(18,22)(19,23)(21,24)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#40164, C₂²×#39282, C₂²×#40056, …
maximal normal subgroup of	C₂²×#40164, C₂²×#39282, C₂²×#40056, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540530);