#7489169

order	512
nilpotency class	3
isoclinism	isoclinic class of #7301076
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	#30038	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1527 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308782, #72308985, #72309037
maximal normal subgroup of	#72308782, #72308985, #72309037

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7396365);