#54430

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	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

696 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)(3,4)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26)(27,32)(28,33)(29,36)(30,31)(34,39)(35,38)(37,40);(1,2,6,5)(3,8,4,7)(10,12)(15,17)(16,20)(22,23)(26,31)(27,35)(28,37)(29,39);(1,3,6,4)(2,7,5,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,30,35)(26,32,31,38)(28,34,37,36)(29,33,39,40);(1,4,6,3)(2,8,5,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,30,37)(26,33,31,40)(27,36,35,34)(29,38,39,32);(1,5,6,2)(3,8,4,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,34)(27,33)(28,38)(30,39)(31,36)(32,37)(35,40)

Polycyclic presentation:

2^8|a^2=fh,b^2=g,c^2=f,d^2=f,e^2=g,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×#54430, #8100793, #8101180, …
maximal normal subgroup of	C₂×#54430, #8100793, #8101180, …
maximal quotient of	#585910, #585916, #587268, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 47418);