C₂×#154635 #72296450

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#154635

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂⁴	C₂⁴×C₄
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

70341 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72278089, C₂×#72278184, C₂×#72278198, …
maximal normal subgroup of	C₂×#72278089, C₂×#72278184, C₂×#72278198, …
maximal quotient of	C₂²×#154635

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

C2×#154635 #72296450