C₂×#726232 #72325122

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

Aliases

direct product

C₂×#726232

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

53177 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232088, C₂×#72232147, C₂×#72232133
maximal normal subgroup of	C₂×#72232088, C₂×#72232147, C₂×#72232133

GAP

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

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

C2×#726232 #72325122