C₂×#7699476 #72298696

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7699476

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

123329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 18 points:

(8,10)(12,17)(15,18);(7,8)(9,10)(11,12,16,17)(13,15,14,18);(4,6)(11,13,16,14)(12,15,17,18);(3,4)(5,6)(11,13,16,14)(12,15,17,18);(11,14,16,13)(12,15,17,18);(11,15)(12,14)(13,17)(16,18);(1,2)

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233771
maximal normal subgroup of	C₂×#72233771
maximal quotient of	C₂×#72298088, C₂×#72298097, C₂×#72298102

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((8,10)(12,17)(15,18), (7,8)(9,10)(11,12,16,17)(13,15,14,18), (4,6)(11,13,16,14)(12,15,17,18), (3,4)(5,6)(11,13,16,14)(12,15,17,18), (11,14,16,13)(12,15,17,18), (11,15)(12,14)(13,17)(16,18), (1,2));

C2×#7699476 #72298696