C₂×#6373939 #72239588

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#6373939

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂³×D₄
commutator	C₂×C₄	C₂⁵×C₄
Frattini, cosocle	C₂²×C₄	C₂⁶
prime radical	C₂²×#3184	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

16345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239592, #72239667, #72239846, …
maximal normal subgroup of	#72239592, #72239667, #72239846, …

GAP

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

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

C2×#6373939 #72239588