#7038717

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1395 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,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, 6945913);