#310494

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10260	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

745 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)(3,6)(4,5)(7,8)(17,18,20,24,21,25,29,22)(19,26,27,30,28,23,31,32);(9,10)(11,13)(12,15)(14,16)(18,22)(20,29)(23,30)(24,25)(26,32)(27,31);(2,5)(6,8)(10,13)(11,16)(14,15)(17,19,21,28)(18,23,25,26)(20,27,29,31)(22,30,24,32);(1,3,4,7)(2,6,5,8)(18,24)(20,29)(22,25)(23,32)(26,30)(27,31)

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 217690);