#7297285

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	#24196	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1499 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,36)(28,38)(29,34)(30,35)(31,32)(33,37)(39,40);(2,5)(3,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,30,31)(26,32,35,36)(28,37,29,39)(33,38,40,34);(1,2)(3,6)(4,5)(7,8)(25,28)(26,33)(27,37)(29,30)(31,39)(32,38)(34,36)(35,40);(2,5)(6,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29,30,28)(26,34,35,38)(27,37,31,39)(32,40,36,33);(1,3,4,7)(2,6,5,8)(10,16)(14,17)(15,24)(22,23)(25,27,30,31)(26,32,35,36)(28,37,29,39)(33,38,40,34)

Polycyclic presentation:

2^9|b^2=f,d^2=f,e^2=hi,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,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282707, #72283145, #72283675
maximal normal subgroup of	#72282707, #72283145, #72283675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7204481);