#57875

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

698 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)(3,7)(4,5)(6,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26)(27,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(2,5)(6,7)(12,20)(15,22)(18,24)(21,23)(25,27,31,29)(26,32,28,35)(30,34,38,37)(33,39,36,40);(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,28,31,26)(27,32,29,35)(30,36,38,33)(34,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,29,31,27)(26,32,28,35)(30,37,38,34)(33,39,36,40);(1,3,4,8)(2,6,5,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,30,31,38)(26,33,28,36)(27,34,29,37)(32,39,35,40)

Polycyclic presentation:

2^8|a^2=gh,b^2=h,c^2=h,d^2=g,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×#57875, #8101488, #8101958, …
maximal normal subgroup of	C₂×#57875, #8101488, #8101958, …
maximal quotient of	#585701, #586382, #586736, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 50863);