#330867

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

492 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)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,18,20,24,21,25,28,23)(19,22,27,31,26,30,32,29)(33,34)(35,38)(36,40)(37,39)(41,45)(42,47)(43,46)(44,48);(2,6)(7,13)(8,14)(15,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32)(33,35)(34,38)(36,41)(37,43)(39,46)(40,45)(42,44)(47,48);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31)(35,41)(36,44)(38,45)(40,48)(42,43)(46,47);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(19,26)(22,30)(27,32)(29,31)(34,39)(38,46)(40,48)(45,47)

Polycyclic presentation:

2^9|a^2=f,c^2=e,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, 238063);