#6532548

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500060
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37546	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6439744);