#290958

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
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	C₂×C₄²	C₂⁴
prime radical	#59213	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=hi,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,g^c=ghi

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, 198154);