#265751

order	512
nilpotency class	3
isoclinism	isoclinic class of #246079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1022 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172947);