#7271936

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fhi,c^2=gi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7179132);