C₂×#203548 #72322166

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#203548

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂⁴	C₂×C₄×D₄
prime radical	C₂⁵	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

4493 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 30 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72318398, C₂×#72318368, C₂×#72320433, …
maximal normal subgroup of	C₂×#72318398, C₂×#72318368, C₂×#72320433, …

GAP

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

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

C2×#203548 #72322166