#328168

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

837 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ei,c^2=i,d^2=i,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, 235364);