#6516642

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1380 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6423838);