#241445

order	512
nilpotency class	3
isoclinism	isoclinic class of #232235
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 148641);