#52883

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	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

647 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)(3,6)(4,7)(5,10)(8,13)(9,11)(12,15)(14,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(36,44)(39,46)(42,48)(45,47);(2,3)(4,11)(8,10)(12,16)(19,26)(22,25)(27,32)(29,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,22)(20,27)(21,26)(23,29)(24,25)(28,32)(30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)

Polycyclic presentation:

2^8|a^2=fg,b^2=gh,c^2=f,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×#52883, #8092562, #8092706, …
maximal normal subgroup of	C₂×#52883, #8092562, #8092706, …
maximal quotient of	#582276, #582466, #582483, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45871);