#438101

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,9,8)(7,11,12,10)(13,14,17,21)(15,24,26,19)(16,28,30,39)(18,23,32,22)(20,33,38,31)(25,37,41,44)(27,35,42,36)(29,40,43,34);(5,7,9,12)(6,10,8,11)(13,15,18,27,17,26,32,42)(14,19,23,36,21,24,22,35)(16,29,31,41,30,43,33,25)(20,37,39,40,38,44,28,34);(13,16,17,30)(14,20,21,38)(15,25,26,41)(18,33,32,31)(19,34,24,40)(22,39,23,28)(27,43,42,29)(35,37,36,44);(2,4)(5,8,9,6)(7,10,12,11)(15,26)(19,24)(25,41)(27,42)(29,43)(34,40)(35,36)(37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345297);