C₂×#519327 #72341718

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

Aliases

direct product

C₂×#519327

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

60489 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:

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

Conjugacy classes & character table

280 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₂²⋊C₄×#525, C₂×#72268519, C₂×#72268523
maximal normal subgroup of	C₂×C₂²⋊C₄×#525, C₂×#72268519, C₂×#72268523

GAP

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

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

C2×#519327 #72341718