#476302

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	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1292 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)(5,6,10,15)(7,16,19,12)(8,13,23,31)(9,11,24,14)(17,29,35,33)(18,26,25,28)(20,27,22,32)(21,36,34,30);(2,4)(6,11)(8,20)(9,24)(12,28)(14,15)(16,26)(17,34)(18,25)(21,35)(22,23)(27,32)(30,36);(1,2)(3,4)(5,7)(6,12)(8,21)(9,25)(10,19)(11,26)(13,30)(14,28)(15,16)(17,22)(18,24)(20,35)(23,34)(27,29)(31,36)(32,33);(2,4)(5,8,10,23)(6,13,15,31)(7,17,19,35)(9,20,24,22)(11,27,14,32)(12,29,16,33)(18,34,25,21)(26,36,28,30)

Polycyclic presentation:

2^9|a^2=f,d^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, 383498);