#7618125

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#31068	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1439 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)(3,8)(4,6)(5,7)(10,14)(11,12)(17,18,21,25)(19,27,28,23)(20,31,33,24)(22,26,34,38)(29,35,41,39)(30,40,42,36)(32,45,46,37)(43,47,44,48);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,16,12,15)(19,28)(20,32)(22,34)(23,27)(24,37)(26,38)(29,43)(31,45)(33,46)(35,47)(39,48)(41,44);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,29)(21,28)(22,30)(24,35)(25,27)(26,36)(31,39)(32,44)(33,41)(34,42)(37,48)(38,40)(43,46)(45,47);(9,11,13,12)(10,15,14,16)(17,20,22,32,21,33,34,46)(18,24,26,37,25,31,38,45)(19,29,30,44,28,41,42,43)(23,35,36,48,27,39,40,47);(2,7)(6,8)(9,12,13,11)(10,15,14,16)(19,28)(20,33)(23,27)(24,31)(30,42)(32,46)(36,40)(37,45)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7525321);