#54444

order	256
nilpotency class	2
isoclinism	isoclinic class of #53390
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:

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

Polycyclic presentation:

2^8|a^2=h,b^2=g,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#54444, #8103643, #8103901, …
maximal normal subgroup of	C₂×#54444, #8103643, #8103901, …
maximal quotient of	#585602, #586930, #587172, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 47432);