C₂×#479961 #72318359

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#479961

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×C₄²
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂³	C₂²×#97
prime radical	C₂²×#41	C₄²
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 42 points:

(1,2);(4,7)(9,10)(11,12,14,19,15,21,27,32,16,22,26,20,28,17,29,31)(13,18,24,34,25,36,39,42,23,33,37,35,38,30,40,41);(3,4,8,9,6,7,5,10)(12,17,22,21)(14,26)(18,30,33,36)(19,32,20,31)(24,37)(27,29)(34,42,35,41)(39,40);(3,5,6,8)(4,9,7,10)(11,13)(12,18)(14,24)(15,25)(16,23)(17,30)(19,34)(20,35)(21,36)(22,33)(26,37)(27,39)(28,38)(29,40)(31,41)(32,42);(13,23)(18,33)(24,37)(25,38)(30,36)(34,35)(39,40)(41,42)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216926, C₂×#72216668, C₂×#72224963, …
maximal normal subgroup of	C₂×#72216926, C₂×#72216668, C₂×#72224963, …

GAP

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

gap> G := Group((1,2), (4,7)(9,10)(11,12,14,19,15,21,27,32,16,22,26,20,28,17,29,31)(13,18,24,34,25,36,39,42,23,33,37,35,38,30,40,41), (3,4,8,9,6,7,5,10)(12,17,22,21)(14,26)(18,30,33,36)(19,32,20,31)(24,37)(27,29)(34,42,35,41)(39,40), (3,5,6,8)(4,9,7,10)(11,13)(12,18)(14,24)(15,25)(16,23)(17,30)(19,34)(20,35)(21,36)(22,33)(26,37)(27,39)(28,38)(29,40)(31,41)(32,42), (13,23)(18,33)(24,37)(25,38)(30,36)(34,35)(39,40)(41,42));

C2×#479961 #72318359