#476310

order	512
nilpotency class	3
isoclinism	isoclinic class of #476112
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	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,g^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,g^b=gi,d^c=de,e^c=ei,g^c=gi,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, 383506);