C₂×#546377 #72324142

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

Aliases

direct product

C₂×#546377

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2433	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

42617 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)(20,23)(24,25);(7,8,9,10)(13,17)(16,18)(19,20)(21,25)(22,23)(24,26);(12,15)(13,17)(21,26)(24,25);(1,2);(3,4)(5,6)(11,12,14,15)(13,16,17,18)(19,20)(21,24)(22,23)(25,26);(11,13,14,17)(12,16,15,18)(19,21,22,26)(20,24,23,25)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297844, C₂×#72262141, C₂×#72323273
maximal normal subgroup of	C₂×#72297844, C₂×#72262141, C₂×#72323273

GAP

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

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

C2×#546377 #72324142