#52882

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

643 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)(3,9,10,6)(4,7,12,14)(11,15,16,13)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(36,44)(39,46)(42,48)(45,47);(3,10)(6,9)(11,16)(13,15)(18,20)(19,26)(24,28)(27,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3)(2,6)(4,11)(5,10)(7,13)(8,9)(12,16)(14,15)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,4,5,12)(2,7,8,14)(3,11,10,16)(6,13,9,15)(17,21,22,29)(18,24,20,28)(19,27,26,32)(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=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×#52882, #8092562, #8092707, …
maximal normal subgroup of	C₂×#52882, #8092562, #8092707, …
maximal quotient of	#582276, #582463, #582485, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45870);