#7523604

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	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2263 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,26)(20,25)(21,27)(22,37)(23,29)(24,28)(30,42)(31,40)(32,39)(33,41)(34,48)(35,46)(36,45)(38,44)(43,47);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,23)(21,34)(22,31)(24,33)(26,29)(27,43)(28,40)(30,35)(32,38)(36,47)(37,41)(39,46)(42,44)(45,48);(10,14)(11,15)(17,20,23,19)(18,26,29,25)(21,32,36,30)(22,31,24,33)(27,42,45,39)(28,40,37,41)(34,35,47,38)(43,46,48,44);(10,14)(13,16)(17,21)(18,27)(19,30)(20,32)(22,35)(23,36)(24,38)(25,39)(26,42)(28,44)(29,45)(31,47)(33,34)(37,46)(40,43)(41,48);(2,6)(5,7)(17,22,23,24)(18,28,29,37)(19,31,20,33)(21,35,36,38)(25,40,26,41)(27,44,45,46)(30,47,32,34)(39,43,42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=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, 7430800);