#6685802

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1722 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22)(23,27)(24,35)(25,28)(26,30)(29,37)(31,46)(32,44)(33,42)(34,40)(36,41)(38,49)(39,43)(45,50)(47,51)(48,52);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,25,33)(22,27,28,42)(24,36,37,32)(26,31,39,34)(29,44,35,41)(30,46,43,40)(38,48,50,47)(45,51,49,52);(1,2,3,4)(22,28)(23,31)(24,37)(26,39)(27,40)(32,48)(33,34)(36,47)(41,52)(42,46)(44,51)(45,49);(6,11)(10,13)(12,19)(18,20)(21,24,25,37)(22,29,28,35)(23,32,33,36)(26,38,39,50)(27,41,42,44)(30,45,43,49)(31,47,34,48)(40,51,46,52);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(22,28)(23,31)(26,39)(27,40)(29,35)(32,47)(33,34)(36,48)(38,50)(41,51)(42,46)(44,52)

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592998);