#402481

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	#14843
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^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, 309677);