#477161

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1145 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384357);