#56823

order	256
nilpotency class	2
isoclinism	isoclinic class of #53390
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

669 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)(3,7)(4,5)(6,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(26,30)(27,33)(28,36)(38,40);(2,5)(6,7)(10,11)(12,19)(13,21)(22,23)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,3,4,8)(2,6,5,7)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^8|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#56823, #8099854, #8100300, …
maximal normal subgroup of	C₂×#56823, #8099854, #8100300, …
maximal quotient of	#585891, #586189, #587021, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 49811);