#7484306

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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

1389 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7391502);