C₂×#648254 #72325418

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

Aliases

direct product

C₂×#648254

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

47723 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232083, C₂×#72232141, C₂×#72232117
maximal normal subgroup of	C₂×#72232083, C₂×#72232141, C₂×#72232117

GAP

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

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

C2×#648254 #72325418