#438980

order	512
nilpotency class	3
isoclinism	isoclinic class of #422540
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

1394 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=f,d^2=e,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^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346176);