#6674579

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(5,6)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22,25,27)(23,31,32,44)(24,35,36,29)(26,39,38,30)(28,40,41,34)(33,47,46,45)(37,43,51,50)(42,48,49,52);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)(22,27)(23,32)(24,36)(30,39)(34,40)(37,51)(42,49)(45,47);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,23)(22,28)(24,33)(25,32)(26,40)(27,41)(29,42)(30,44)(31,39)(34,38)(35,49)(36,46)(37,52)(43,47)(45,50)(48,51);(6,13)(11,15)(12,14)(19,20)(21,24,25,36)(22,29,27,35)(23,33,32,46)(26,37,38,51)(28,42,41,49)(30,43,39,50)(31,45,44,47)(34,48,40,52);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(23,32)(24,36)(28,41)(29,35)(31,44)(34,40)(37,51)(43,50)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6581775);