#249200

order	512
nilpotency class	3
isoclinism	isoclinic class of #249199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#36717	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1625 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,d^b=di,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156396);