#7558910

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=fi,d^2=gi,e^2=fi,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,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7466106);