C₂×#183596 #72321822

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

Aliases

direct product

C₂×#183596

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂⁴	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5237 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)(10,11,14,13)(15,16,19,22,20,23,24,17)(18,21,27,29,26,28,30,25);(8,11)(10,14)(12,13)(15,17,24,23,20,22,19,16)(18,25,30,28,26,29,27,21);(15,18)(16,21)(17,25)(19,27)(20,26)(22,29)(23,28)(24,30);(3,4,5,6)(18,26)(21,28)(25,29)(27,30)

Conjugacy classes & character table

280 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72318410, C₂×#72318402, C₂×#72318419, …
maximal normal subgroup of	C₂×#72318410, C₂×#72318402, C₂×#72318419, …

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

C2×#183596 #72321822