#52845

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	C₂×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^8|a^2=f,b^2=fgh,c^2=g,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×#52845, #8092651, #8092705, …
maximal normal subgroup of	C₂×#52845, #8092651, #8092705, …
maximal quotient of	#582318, #582504, #582543, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45833);