C₂×#845523 #72332871

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#845523

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2847	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

38059 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)(11,12,15,13)(14,17,18,16)(20,24)(22,25);(7,8)(9,10)(12,13)(16,17)(19,20)(21,25)(22,26)(23,24);(19,21)(20,22)(23,26)(24,25);(1,2);(4,6)(11,13,15,12)(14,17,18,16)(19,21)(20,25)(22,24)(23,26);(3,4)(5,6)(11,14,15,18)(12,16,13,17)(19,22,23,25)(20,26,24,21)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323396, C₂×#72323390, C₂×#72323403
maximal normal subgroup of	C₂×#72323396, C₂×#72323390, C₂×#72323403

GAP

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

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

C2×#845523 #72332871