#6672105

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1693 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:

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

Polycyclic presentation:

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

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, 6579301);