C₂×C₂₇² #13089928

order	1458
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 314928
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂₇

Aliases

abelian

C₂×C₂₇²

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉²	C₂×C₃²
socle, prime radical	C₂×C₃²	C₉²
LCS	G ▷ C₁
UCS	C₁ ◁ G

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 1458

A minimal faithful permutation representation — 2 generators on 56 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29);(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)

Conjugacy classes & character table

1458 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×C₂₇², C₂²×C₂₇², C₂×C₂₇×C₈₁, …
maximal normal subgroup of	C₄×C₂₇², C₂²×C₂₇², C₂×C₂₇×C₈₁, …
maximal quotient of	C₄×C₂₇², C₂²×C₂₇², C₂×C₂₇×C₈₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1458, 2);

C2×C272 #13089928