#6370366

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=f,g^2=i,b^a=bf,d^a=df,c^b=cf,e^b=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277562);