#476301

order	512
nilpotency class	3
isoclinism	isoclinic class of #476106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1340 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:

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

Polycyclic presentation:

2^9|a^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383497);