C₂×#113024 #72233682

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#113024

Subgroups

	subgroup	G/H
center, socle	C₂³	#1438
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	C₂×#60558	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

7591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72213800, C₂×#72213848, C₂×#72215564, …
maximal normal subgroup of	C₂×#72213800, C₂×#72213848, C₂×#72215564, …

GAP

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

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

C2×#113024 #72233682