#7170233

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1436 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)(10,13)(11,15)(17,18,23,35,21,24,22,27)(19,28,34,25,30,41,31,37)(20,26,36,46,32,38,33,40)(29,42,45,39,43,48,44,47);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(18,24)(20,32)(25,37)(27,35)(28,41)(29,43)(33,36)(44,45);(9,10)(11,14)(12,13)(15,16)(17,19,21,30)(18,25,24,37)(20,29,32,43)(22,34,23,31)(26,39,38,47)(27,41,35,28)(33,45,36,44)(40,48,46,42);(3,8)(5,7)(10,13)(14,16)(17,20,21,32)(18,26,24,38)(19,29,30,43)(22,33,23,36)(25,39,37,47)(27,40,35,46)(28,42,41,48)(31,44,34,45);(9,11,12,15)(10,14,13,16)(20,32)(26,38)(29,43)(33,36)(39,47)(40,46)(42,48)(44,45)

Polycyclic presentation:

2^9|a^2=fgh,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7077429);