C₂×#183475 #72321885

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#183475

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂⁴	C₂×C₄×D₄
prime radical	C₂⁵	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5365 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72318410, C₂×#72318368, C₂×#72320446
maximal normal subgroup of	C₂×#72318410, C₂×#72318368, C₂×#72320446

GAP

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

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

C2×#183475 #72321885