#52787

order	256
nilpotency class	2
isoclinism	isoclinic class of #52308
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

625 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:

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

Polycyclic presentation:

2^8|a^2=g,b^2=gh,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#52787, #8092543, #8092651, …
maximal normal subgroup of	C₂×#52787, #8092543, #8092651, …
maximal quotient of	#582332, #582497, #582564, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45775);