#293128

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
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	#24658	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

925 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(17,18,22,25)(19,27,30,24)(20,21,26,23)(28,32,31,29);(1,3,4,8)(2,5,6,7)(10,13)(11,16)(14,15)(17,19)(18,24)(20,31)(21,32)(22,30)(23,29)(25,27)(26,28);(9,11,13,16,12,15,10,14)(17,20,23,18,22,26,21,25)(19,28,29,27,30,31,32,24);(2,6)(5,7)(9,10,12,13)(11,16,15,14)(17,21,22,23)(18,20,25,26)(19,29,30,32)(24,28,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 200324);