#418511

order	512
nilpotency class	3
isoclinism	isoclinic class of #418510
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15263
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1675 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)(3,11)(4,18)(5,24)(6,13)(7,12)(8,14)(9,16)(10,21)(15,25)(17,30)(19,26)(20,27)(22,29)(23,28)(31,32);(1,3)(2,9)(4,19)(5,6)(7,16)(8,17)(10,26)(11,13)(12,30)(14,24)(15,21)(18,32)(20,23)(22,31)(25,28)(27,29);(1,4)(2,10)(3,15)(5,20)(6,21)(7,22)(8,23)(9,25)(11,27)(12,18)(13,28)(14,29)(16,19)(17,31)(24,32)(26,30);(1,5,7,17,8,16,6,3)(2,11,13,9,14,30,12,24)(4,20,22,31,23,19,21,15)(10,27,28,25,29,26,18,32)

Polycyclic presentation:

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

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, 325707);