#41704

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

697 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)(9,10)(11,17)(12,20)(13,14)(15,19)(16,22)(18,23)(21,24)(25,26,30,31)(27,32,34,29)(28,36,38,33)(35,37,40,39);(10,14)(15,17)(16,20)(23,24)(26,31)(27,34)(33,36)(35,40);(1,2,4,5)(3,6,7,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40)

Polycyclic presentation:

2^8|a^2=h,c^2=g,d^2=f,e^2=fg,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×#41704, #7640071, #7640237, …
maximal normal subgroup of	C₂×#41704, #7640071, #7640237, …
maximal quotient of	#544845, #545272, #545803, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34692);