#7523596

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,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, 7430792);