#6498614

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469956
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1816 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)(21,22,26,29,27,30,25,35)(23,31,33,24,32,34,36,28);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,27,32)(22,28,30,24)(25,33,26,36)(29,34,35,31);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,24,27,28)(22,23,30,32)(25,31,26,34)(29,33,35,36);(7,13)(10,17)(14,19)(18,20)(23,32)(24,28)(31,34)(33,36);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(21,25,27,26)(22,29,30,35)(23,33,32,36)(24,31,28,34)

Polycyclic presentation:

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

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, 6405810);