#6497734

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1910 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6404930);