#304833

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 212029);