#312825

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=ei,d^2=ef,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,d^c=di,g^c=gi

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, 220021);