#349759

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
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

826 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ei,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^b=di,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256955);