C₂×#544514 #72339416

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#544514

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

54205 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297845, C₂×#72327589, C₂×#72327626
maximal normal subgroup of	C₂×#72297845, C₂×#72327589, C₂×#72327626

GAP

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

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

C2×#544514 #72339416