#6477574

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	#37273	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3025 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)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(22,28)(23,32)(25,26)(29,36)(30,31)(34,35);(2,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,22,27,31)(23,33,29,24)(25,28,26,30)(32,34,36,35);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,23,27,29)(22,24,31,33)(25,32,26,36)(28,35,30,34);(7,14)(10,18)(13,15)(17,19)(21,24)(22,29)(23,31)(25,35)(26,34)(27,33)(28,36)(30,32);(6,12)(10,18)(11,20)(17,19)(21,25,27,26)(22,30,31,28)(23,32,29,36)(24,34,33,35)

Polycyclic presentation:

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

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, 6384770);