#54470

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

650 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:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=fg,c^2=f,e^2=fg,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×#54470, #8101887, #8101960, …
maximal normal subgroup of	C₂×#54470, #8101887, #8101960, …
maximal quotient of	#586680, #586758, #587124, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 47458);