#7617845

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1249 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7525041);