#21357

order	256
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	Aut(G) = C₂¹³ (order 8192) · Inn(G) = C₂⁴ · Out(G) = C₂⁹
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

415 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^8|a^2=g,b^2=g,d^2=f,g^2=h,b^a=be,c^a=cf,d^c=dh

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#21357, #6372287, #6372290, …
maximal normal subgroup of	C₂×#21357, #6372287, #6372290, …
maximal quotient of	#207388, #207389, #207390, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 14345);