#476303

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
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1284 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)(9,24)(10,14)(11,18)(13,25)(16,32)(17,28)(19,27)(20,35)(21,30)(22,29)(23,31)(26,34)(33,36);(2,4)(5,7,10,19)(6,11,14,28)(8,20,23,16)(9,17,25,18)(12,26,31,36)(13,27,24,15)(21,34,22,33)(29,35,30,32);(1,2)(3,4)(5,8)(6,12)(7,16)(9,21)(10,23)(11,26)(13,29)(14,31)(15,32)(17,33)(18,34)(19,20)(22,25)(24,30)(27,35)(28,36);(2,4)(7,17)(8,21)(9,25)(11,27)(12,29)(13,24)(15,28)(18,19)(22,23)(30,31)(32,35)(33,34)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 383499);