#297003

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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	#39378	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

915 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=f,d^2=ehi,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=dhi

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, 204199);