#6496409

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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)(21,22,26,29)(23,27,32,43)(24,34,37,46)(25,38,40,30)(28,39,45,36)(31,47,35,52)(33,50,49,42)(41,51,44,48);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,27)(24,35)(25,33)(26,32)(28,44)(29,43)(30,42)(31,37)(34,52)(36,51)(38,50)(39,48)(40,49)(41,45)(46,47);(7,13)(10,17)(14,19)(18,20)(21,24,26,37)(22,28,29,45)(23,31,32,35)(25,39,40,36)(27,41,43,44)(30,46,38,34)(33,51,49,48)(42,52,50,47);(7,13)(10,17)(14,19)(18,20);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,29)(23,32)(28,45)(30,38)(31,35)(33,49)(34,46)(48,51)

Polycyclic presentation:

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

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, 6403605);