C₂×#8544 #188030

order	512
nilpotency class	3
isoclinism	isoclinic class of #8541
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#8544

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

4274 conjugacy classes of subgroups; only the 354 normal subgroups are shown.

normal subgroups only

This group has 4274 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)(5,9,6,8)(7,10)(11,12)(13,19,21,16)(14,17)(15,18)(20,25,26,23)(22,24);(3,5,7,6)(4,8,10,9)(11,13)(12,16)(14,20)(15,21)(17,23)(18,19)(22,26)(24,25);(3,6,7,5)(4,9,10,8)(11,14,15,22)(12,17,18,24)(13,20,21,26)(16,23,19,25)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266477, C₂×#188032, C₂×#133855
maximal normal subgroup of	#72266477, C₂×#188032, C₂×#133855
maximal quotient of	C₂×#122049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95226);