#7409968

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	#30414	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1611 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:

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

Polycyclic presentation:

2^9|a^2=fh,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7317164);