#22557

order	256
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	Aut(G) = C₂¹³ (order 8192) · Inn(G) = C₂⁴ · Out(G) = C₂⁹
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator, socle	C₂³	C₂³×C₄
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

495 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#22557, #6379653, #6379655, …
maximal normal subgroup of	C₂×#22557, #6379653, #6379655, …
maximal quotient of	#225066, #225067, #225068, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 15545);