#198751

order	512
nilpotency class	3
isoclinism	isoclinic class of #198516
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60750	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105947);