C₂×#512676 #72292713

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#512676

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2127
commutator	C₄²	C₂⁴×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁵
prime radical	C₂×#60808	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂×#3449 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

5755 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)(7,21,24,12)(9,22)(10,23,19,20)(14,27)(15,31)(16,25)(26,30)(29,32)(33,34);(3,5,6,15,8,14,18,30)(4,9,10,26,11,25,19,31)(7,22,23,32,24,16,20,33)(12,13,28,29,21,27,17,34);(5,14)(6,18)(9,25)(10,19)(13,27)(16,22)(17,28)(20,23);(3,4)(5,9)(6,19)(7,12)(8,11)(10,18)(13,22)(14,25)(15,31)(16,27)(17,23)(20,28)(21,24)(26,30)(29,33)(32,34)

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72292692, C₂×#72292696
maximal normal subgroup of	C₂×#72292692, C₂×#72292696
maximal quotient of	#72293204, #72293212

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

C2×#512676 #72292713