#329458

order	512
nilpotency class	3
isoclinism	isoclinic class of #329455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

820 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)(17,18,22,27,23,28,21,26)(19,24,29,25,20,30,31,32);(2,5)(6,8)(9,10)(11,13)(12,15)(14,16)(17,19)(18,24)(20,23)(21,31)(22,29)(25,27)(26,32)(28,30);(1,3)(2,6)(4,7)(5,8)(10,13)(11,16)(14,15)(17,20,23,19)(18,25,28,32)(21,31,22,29)(24,26,30,27);(3,7)(6,8)(17,21,23,22)(18,26,28,27)(19,29,20,31)(24,25,30,32)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,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, 236654);