#327754

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

897 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)(3,6)(4,5)(7,8)(9,10,13,18)(11,19,21,15)(12,17,14,16)(20,24,22,23)(25,26)(27,32)(28,30)(29,33)(31,39)(34,35)(36,40)(37,38);(2,5)(6,8)(9,11,13,21)(10,15,18,19)(12,22,14,20)(16,23,17,24)(25,27)(26,32)(28,38)(29,36)(30,37)(31,35)(33,40)(34,39);(1,3)(2,6)(4,7)(5,8)(10,16)(12,14)(15,23)(17,18)(19,24)(20,22)(25,28,30,26)(27,34,37,40)(29,39,31,33)(32,35,38,36);(3,7)(6,8)(9,12,13,14)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,30,31)(26,33,28,39)(27,35,37,36)(32,34,38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dhi,f^b=fi,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234950);