#6628976

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628872
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
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4459 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)(5,6,11,9)(7,12,17,16)(8,19,22,28)(10,23,25,14)(13,24,20,21)(15,29,31,36)(18,33,34,27)(26,35,30,32);(1,2,3,4)(5,7)(6,12)(8,15)(9,16)(10,18)(11,17)(13,26)(14,27)(19,29)(20,30)(21,32)(22,31)(23,33)(24,35)(25,34)(28,36);(5,8)(6,13)(7,15)(9,20)(10,24)(11,22)(12,26)(14,28)(16,30)(17,31)(18,35)(19,23)(21,25)(27,36)(29,33)(32,34);(5,9,11,6)(7,16,17,12)(8,20,22,13)(10,14,25,23)(15,30,31,26)(18,27,34,33)(19,24,28,21)(29,35,36,32);(7,17)(12,16)(15,31)(18,34)(26,30)(27,33)(29,36)(32,35)

Polycyclic presentation:

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

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, 6536172);