#345628

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

585 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=ehi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252824);