#330624

order	512
nilpotency class	3
isoclinism	isoclinic class of #329647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

524 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237820);