#6524796

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3562	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2958 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)(5,6)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22,26,30,27,23,25,29)(24,32,33,31,34,36,35,28);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,33)(35,36);(6,11)(7,13)(8,15)(16,20)(21,24)(22,28)(23,31)(25,33)(26,35)(27,34)(29,32)(30,36);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,16,17,20)(21,25,27,26)(22,29,23,30)(24,33,34,35)(28,32,31,36);(7,13)(10,17)(14,19)(16,20)(21,25,27,26)(22,30,23,29)(24,33,34,35)(28,36,31,32)

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6431992);