#330858

order	512
nilpotency class	3
isoclinism	isoclinic class of #329587
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

573 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=ehi,d^2=e,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, 238054);