#6498370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470922
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	#37559	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22)(23,32)(24,34)(25,30)(26,31)(27,33)(28,29)(35,36);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23)(22,28)(24,33)(25,34)(26,29)(27,35)(30,36)(31,32);(6,11)(7,13)(12,15)(14,19)(21,24,27,28,26,36,25,32)(22,29,30,34,31,23,33,35);(7,13)(10,17)(14,19)(18,20)(21,25,26,27)(22,30,31,33)(23,34,29,35)(24,28,36,32);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(23,29)(24,36)(28,32)(34,35)

Polycyclic presentation:

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

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, 6405566);