#6684545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1773 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei

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, 6591741);