#402884

order	512
nilpotency class	3
isoclinism	isoclinic class of #394482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2435
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14860
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310080);