#6442572

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437452
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349768);