C₂×#483311 #72292771

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

Aliases

direct product

C₂×#483311

Subgroups

	subgroup	G/H
center, socle	C₂²	#17754
commutator	C₄²	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	C₂×#60704	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#2873 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72292716, C₂×#72292717
maximal normal subgroup of	C₂×#72292716, C₂×#72292717

GAP

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

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

C2×#483311 #72292771