#7523983

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,32)(21,25)(22,26)(23,30)(24,34)(28,39)(29,40)(31,45)(33,42)(35,46)(36,44)(37,41)(38,43)(47,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,30)(18,23,25,27)(20,33,34,28)(22,37,36,31)(24,42,32,39)(26,45,44,41)(29,38,48,35)(40,46,47,43);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,21,34)(18,24,25,32)(19,28,30,33)(22,35,36,38)(23,39,27,42)(26,43,44,46)(29,37,48,31)(40,45,47,41);(10,14)(13,16)(17,20)(18,24)(19,29)(21,34)(22,38)(23,40)(25,32)(26,46)(27,47)(28,31)(30,48)(33,37)(35,36)(39,41)(42,45)(43,44);(2,5)(6,8)(18,25)(20,34)(23,27)(26,44)(28,33)(29,48)(35,38)(41,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7431179);