#330860

order	512
nilpotency class	3
isoclinism	isoclinic class of #329587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

521 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)(17,18,22,30,20,23,21,25)(19,26,29,24,27,32,28,31)(33,34)(35,42)(36,46)(37,41)(38,40)(39,45)(43,48)(44,47);(2,6)(7,13)(8,14)(15,16)(18,23)(24,31)(25,30)(26,32)(33,35)(34,39)(36,44)(37,45)(38,43)(40,47)(41,42)(46,48);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,19,20,27)(18,24,23,31)(21,29,22,28)(25,32,30,26)(35,43)(36,38)(39,47)(40,46)(42,48)(44,45);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(19,27)(24,31)(26,32)(28,29)(33,36,37,38)(34,40,41,46)(35,44,45,43)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=fh,c^2=ei,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238056);