#53363

order	256
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	Aut(G) = C₂¹⁵ (order 32768) · Inn(G) = C₂⁵ · Out(G) = C₂¹⁰
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

627 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^8|a^2=f,b^2=g,c^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#53363, #8105918, #8105920, …
maximal normal subgroup of	C₂×#53363, #8105918, #8105920, …
maximal quotient of	#606831, #606833, #606843, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 46351);