#6526772

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499814
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,30)(24,29)(25,33)(26,28)(27,31)(32,36)(34,35);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,27)(24,31)(25,34)(26,32)(28,35)(29,36)(30,33);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(21,22)(23,27)(24,33)(25,29)(26,28)(30,31)(32,35)(34,36);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(22,28)(27,35)(29,33)(30,36);(7,13)(10,16)(14,19)(17,20)(21,24,26,25)(22,29,28,33)(23,31,32,34)(27,36,35,30)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,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, 6433968);