#421766

order	512
nilpotency class	3
isoclinism	isoclinic class of #418871
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2697
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15297
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1492 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328962);