C₂×#481317 #72318604

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#481317

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂³	C₂²×#102
prime radical	C₂×#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5453 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72216916, C₂×#72216906, C₂×#72216701, …
maximal normal subgroup of	C₂×#72216916, C₂×#72216906, C₂×#72216701, …

GAP

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

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

C2×#481317 #72318604