#41795

order	256
nilpotency class	2
isoclinism	isoclinic class of #40484
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

695 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:

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

Polycyclic presentation:

2^8|a^2=gh,b^2=f,c^2=f,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#41795, #7639714, #7640160, …
maximal normal subgroup of	C₂×#41795, #7639714, #7640160, …
maximal quotient of	#544889, #545298, #545887, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34783);