C₂×#2315887 #72332648

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#2315887

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

32301 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323396, C₂×#72323408, C₂×#72323387
maximal normal subgroup of	C₂×#72323396, C₂×#72323408, C₂×#72323387

GAP

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

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

C2×#2315887 #72332648