#466728

order	512
nilpotency class	3
isoclinism	isoclinic class of #457090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1320 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:

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

Polycyclic presentation:

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

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, 373924);