#6497515

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469932
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 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)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,30)(24,28)(25,33)(26,29)(27,31)(32,35)(34,36);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,27)(24,31)(25,34)(26,32)(28,35)(29,36)(30,33);(1,2,3,4)(7,13)(10,17)(14,19)(18,20);(7,13)(10,17)(14,19)(18,20)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6404711);