C₂×#20772 #6371062

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#20772

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3234 conjugacy classes of subgroups; only the 1282 normal subgroups are shown.

normal subgroups only

This group has 3234 conjugacy classes of subgroups — too many to draw. Only its 1282 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) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257397, #72257407, #72257459, …
maximal normal subgroup of	#72257397, #72257407, #72257459, …
maximal quotient of	#72265353

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278258);