#41791

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	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

677 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)(9,10,14,12)(11,17,18,15)(13,16,21,20)(19,23,24,22)(25,26,31,30)(27,34,28,32)(29,37,38,33)(35,39,36,40);(11,18)(13,21)(15,17)(16,20)(25,27,31,28)(26,32,30,34)(29,35,38,36)(33,39,37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,28,31,27)(26,32,30,34)(29,35,38,36)(33,40,37,39);(1,3,5,4)(2,7,6,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,29,31,38)(26,33,30,37)(27,35,28,36)(32,39,34,40);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,30,31,26)(27,32,28,34)(29,37,38,33)(35,39,36,40)

Polycyclic presentation:

2^8|a^2=fh,b^2=f,c^2=f,d^2=fg,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×#41791, #7639714, #7640222, …
maximal normal subgroup of	C₂×#41791, #7639714, #7640222, …
maximal quotient of	#544993, #545352, #545903, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34779);