#7170524

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	#42652	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1579 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)(17,18,22,27,21,26,35,24)(19,28,31,39,29,23,36,37)(20,25,34,42,33,43,46,40)(30,41,44,48,32,38,45,47);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,23)(20,32)(21,29)(22,31)(24,39)(25,41)(26,28)(27,37)(30,33)(34,45)(35,36)(38,43)(40,47)(42,48)(44,46);(9,10)(11,14)(12,13)(15,16)(18,24)(19,29)(20,33)(22,35)(23,37)(25,42)(26,27)(28,39)(38,48)(40,43)(41,47)(44,45);(3,8)(5,7)(10,13)(14,16)(17,20)(18,25)(19,30)(21,33)(22,34)(23,38)(24,40)(26,43)(27,42)(28,41)(29,32)(31,44)(35,46)(36,45)(37,47)(39,48);(9,11,12,15)(10,14,13,16)(18,26)(19,29)(20,33)(24,27)(31,36)(34,46)(38,41)(47,48)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,e^2=hi,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, 7077720);