#57715

order	256
nilpotency class	2
isoclinism	isoclinic class of #53390
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	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

645 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)(3,7,9,11)(5,12,13,8)(10,16,15,14)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,34)(35,40)(36,42)(37,39)(38,41)(43,47)(44,48)(45,46);(2,4)(3,9)(5,13)(14,16)(18,21)(20,29)(23,25)(26,30)(33,35,39,36)(34,40,37,42)(38,44,46,43)(41,48,45,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19)(18,23)(20,30)(21,25)(22,27)(24,32)(26,29)(28,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,32)(27,30)(33,37,39,34)(35,42,36,40)(38,45,46,41)(43,48,44,47);(1,5,6,13)(2,8,4,12)(3,10,9,15)(7,14,11,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,32,30,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^8|a^2=h,b^2=fgh,c^2=fgh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#57715, #8101833, #8101951, …
maximal normal subgroup of	C₂×#57715, #8101833, #8101951, …
maximal quotient of	#586264, #586422, #586461, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 50703);