C₂×#612965 #72299355

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#612965

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2537	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

29385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297597, C₂×#72297594, C₂×#72297592
maximal normal subgroup of	C₂×#72297597, C₂×#72297594, C₂×#72297592

GAP

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

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

C2×#612965 #72299355