#6498632

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469989
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	#35303	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1841 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=gh,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6405828);