C₂×#72232051 #72324475

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰×C₄

Aliases

direct product

C₂×#72232051

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

709528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72232064
maximal normal subgroup of	C₂×#72232064

GAP

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

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

C2×#72232051 #72324475