#418580

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1486 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=dei,f^c=fi,h^c=hi,f^d=fi

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, 325776);