#47526

order	256
nilpotency class	2
isoclinism	isoclinic class of #46350
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

688 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:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=f,c^2=fgh,d^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#47526, #7700944, #7700968, …
maximal normal subgroup of	C₂×#47526, #7700944, #7700968, …
maximal quotient of	#576828, #576851, #577070, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 40514);