#327868

order	512
nilpotency class	3
isoclinism	isoclinic class of #326783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

752 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,48)(44,46)(45,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(33,35,38,45)(34,39,42,47)(36,43,37,44)(40,48,41,46);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25)(33,36,38,37)(34,40,42,41)(35,43,45,44)(39,46,47,48);(4,12)(7,14)(10,16)(13,15)(18,23)(22,31)(24,28)(25,32)(33,37,38,36)(34,41,42,40)(35,44,45,43)(39,46,47,48)

Polycyclic presentation:

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

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, 235064);