#329997

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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

576 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=i,d^2=i,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, 237193);