C₂×#615419 #72331934

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

Aliases

direct product

C₂×#615419

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

45041 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)(14,18)(16,17);(4,6)(7,8)(9,10)(11,12,15,13)(14,16,18,17)(19,20,22,24)(21,25,26,23);(19,20)(21,23)(22,24)(25,26);(1,2);(3,4,5,6)(11,13,15,12)(14,17,18,16)(19,20)(21,23)(22,24)(25,26);(11,14)(12,16)(13,17)(15,18)(19,21,22,26)(20,23,24,25)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323316, C₂×#72323310, C₂×#72323330
maximal normal subgroup of	C₂×#72323316, C₂×#72323310, C₂×#72323330

GAP

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

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

C2×#615419 #72331934