#330608

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

528 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)(27,30)(28,32)(29,31)(33,37)(34,39)(35,38)(36,40);(2,5)(6,8)(9,11)(10,16)(12,19)(13,22)(14,21)(15,20)(17,24)(18,23)(25,27)(26,30)(28,33)(29,35)(31,38)(32,37)(34,36)(39,40);(3,7)(6,8)(9,12,14,10)(11,16,21,19)(13,17,15,18)(20,24,22,23)(27,33)(28,36)(30,37)(32,40)(34,35)(38,39);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,17,12,18)(11,20,21,22)(16,23,19,24)(26,31)(30,38)(32,40)(37,39)

Polycyclic presentation:

2^9|a^2=ef,c^2=h,d^2=e,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, 237804);