#302996

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ehi,c^2=f,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210192);