C₂×#525951 #72325285

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

Aliases

direct product

C₂×#525951

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

54901 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232087, C₂×#72232121, C₂×#72232166
maximal normal subgroup of	C₂×#72232087, C₂×#72232121, C₂×#72232166

GAP

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

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

C2×#525951 #72325285