#442125

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efg,c^2=ef,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, 349321);