#395290

order	512
nilpotency class	3
isoclinism	isoclinic class of #393963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 302486);