C₂×#618455 #72332898

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#618455

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

70821 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323386, C₂×#72323393, C₂×#72237461, …
maximal normal subgroup of	C₂×#72323386, C₂×#72323393, C₂×#72237461, …
maximal quotient of	C₂×#72323614, #72339976

GAP

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

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

C2×#618455 #72332898