C₂×#855329 #72325327

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#855329

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

54403 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232188, C₂×#72232156, C₂×#72232159
maximal normal subgroup of	C₂×#72232188, C₂×#72232156, C₂×#72232159

GAP

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

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

C2×#855329 #72325327