#7590511

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 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:

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

Polycyclic presentation:

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

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, 7497707);