#327672

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

932 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)(9,10,12,14)(11,16,13,15)(17,18)(19,24)(20,22)(21,25)(23,31)(26,27)(28,32)(29,30);(2,5)(6,8)(10,14)(15,16)(17,19,22,29)(18,24,20,30)(21,28,23,27)(25,32,31,26);(1,3)(2,6)(4,7)(5,8)(10,15)(11,13)(14,16)(17,20,22,18)(19,26,29,32)(21,31,23,25)(24,27,30,28);(3,7)(6,8)(9,11,12,13)(10,16,14,15)(17,21,22,23)(18,25,20,31)(19,27,29,28)(24,26,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234868);