#46220

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

701 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:

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

Polycyclic presentation:

2^8|a^2=h,c^2=fg,d^2=fgh,e^2=g,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×#46220, #7702765, #7702852, …
maximal normal subgroup of	C₂×#46220, #7702765, #7702852, …
maximal quotient of	#557180, #557220, #557485, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39208);