C₂×#612801 #72278633

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂×#612801

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

440729 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232180, C₂×#72231456, C₂×D₄×#2846, …
maximal normal subgroup of	C₂×#72232180, C₂×#72231456, C₂×D₄×#2846, …
maximal quotient of	#72286712, C₂²×#612801, C₂×#72286393, …

GAP

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

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

C2×#612801 #72278633