C₂×#535282 #72325049

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#535282

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#534	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

65417 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72219261, C₂×#72232118, C₂×#72232144
maximal normal subgroup of	C₂×#72219261, C₂×#72232118, C₂×#72232144

GAP

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

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

C2×#535282 #72325049