#245909

order	512
nilpotency class	3
isoclinism	isoclinic class of #245903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1229 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153105);