#46304

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

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:

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

Polycyclic presentation:

2^8|b^2=f,c^2=fg,d^2=gh,e^2=h,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×#46304, #7702975, #7703002, …
maximal normal subgroup of	C₂×#46304, #7702975, #7703002, …
maximal quotient of	#557299, #557514, #557521, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39292);