#52864

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

629 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,25,27,20)(21,24,29,30)(26,31,32,28)(36,44)(39,46)(42,48)(45,47);(2,4)(3,9)(8,12)(11,15)(18,23)(20,25)(24,30)(28,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3,6,9)(2,7,4,10)(5,11,13,15)(8,14,12,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,31)(30,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,21,22,29)(18,24,23,30)(19,26,27,32)(20,28,25,31)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)

Polycyclic presentation:

2^8|a^2=g,b^2=fgh,c^2=fg,d^2=fg,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×#52864, #8092603, #8092704, …
maximal normal subgroup of	C₂×#52864, #8092603, #8092704, …
maximal quotient of	#582350, #582499, #582565, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45852);