C₂×#8552 #188079

order	512
nilpotency class	3
isoclinism	isoclinic class of #8541
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#8552

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

4102 conjugacy classes of subgroups; only the 354 normal subgroups are shown.

normal subgroups only

This group has 4102 conjugacy classes of subgroups — too many to draw. Only its 354 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) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#188085, #72238534, #72261622
maximal normal subgroup of	C₂×#188085, #72238534, #72261622

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95275);