#52894

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:

(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(36,44)(39,46)(42,48)(45,47);(18,20)(23,25)(24,28)(30,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,13)(14,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(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=h,c^2=fg,d^2=f,e^2=g,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×#52894, #8092562, #8092714, …
maximal normal subgroup of	C₂×#52894, #8092562, #8092714, …
maximal quotient of	#582285, #582473, #582481, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45882);