C₂×#6373306 #72239628

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#6373306

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂³×D₄
commutator	C₂×C₄	C₂⁵×C₄
Frattini, cosocle	C₂²×C₄	C₂⁶
prime radical	C₂²×#3109	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

18785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239667
maximal normal subgroup of	#72239667

GAP

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

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

C2×#6373306 #72239628