C₂×#157600 #72294948

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#157600

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

63893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72278695, C₂×#72279098, C₂×#72293962, …
maximal normal subgroup of	C₂×#72278695, C₂×#72279098, C₂×#72293962, …

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

C2×#157600 #72294948