#442127

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

897 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)(5,6,10,17,12,18,11,16)(7,19,29,34,22,33,21,13)(8,23,30,14,26,32,25,35)(9,27,28,15,24,36,20,31);(5,7,10,29,12,22,11,21)(6,13,17,19,18,34,16,33)(8,24,30,20,26,9,25,28)(14,27,32,15,35,36,23,31);(5,8)(6,14,18,35)(7,20)(9,21)(10,25)(11,30)(12,26)(13,31,34,15)(16,32,17,23)(19,36,33,27)(22,28)(24,29);(2,4)(5,9,10,28,12,24,11,20)(6,15,16,27,18,31,17,36)(7,8,21,25,22,26,29,30)(13,32,19,35,34,23,33,14)

Polycyclic presentation:

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

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, 349323);