#328053

order	512
nilpotency class	3
isoclinism	isoclinic class of #327117
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

688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235249);