#45879

order	256
nilpotency class	2
isoclinism	isoclinic class of #45438
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

696 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)(6,8)(10,14)(11,17)(16,21)(18,23)(25,26,30,33)(27,28,35,31)(29,37,39,32)(34,40,38,36);(10,14)(11,17)(12,19)(22,24)(25,27,30,35)(26,31,33,28)(29,38,39,34)(32,36,37,40);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,3,4,7)(2,6,5,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,39)(26,32,33,37)(27,34,35,38)(28,36,31,40)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#45879, #7702451, #7702509, …
maximal normal subgroup of	C₂×#45879, #7702451, #7702509, …
maximal quotient of	#557264, #557341, #557426, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 38867);