#41549

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

Polycyclic presentation:

2^8|a^2=fh,c^2=fg,d^2=fg,e^2=f,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×#41549, #7640139, #7640229, …
maximal normal subgroup of	C₂×#41549, #7640139, #7640229, …
maximal quotient of	#544673, #545189, #545724, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34537);