#329453

order	512
nilpotency class	3
isoclinism	isoclinic class of #329395
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	#32904	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

807 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)(3,6)(4,5)(7,8)(9,10,11,15,12,14,13,16)(17,18,20,24,21,25,29,23)(19,22,26,30,28,32,27,31);(2,5)(6,8)(10,14)(15,16)(17,19)(18,22)(20,26)(21,28)(23,31)(24,30)(25,32)(27,29);(1,3)(2,6)(4,7)(5,8)(10,15)(11,13)(14,16)(18,23)(19,26)(20,29)(24,25)(27,28)(30,31);(3,7)(6,8)(9,11,12,13)(10,15,14,16)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236649);