#326054

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
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

608 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)(3,8)(4,6)(5,7)(17,18,22,26)(19,24,29,32)(20,30,28,23)(21,31,27,25);(1,3,5,4)(2,6,7,8)(9,10,13,14,12,15,11,16)(17,19,27,20,22,29,21,28)(18,23,31,32,26,30,25,24);(1,4,5,3)(2,6,7,8)(10,14)(11,13)(15,16)(17,20,21,19,22,28,27,29)(18,24,25,30,26,32,31,23);(2,7)(6,8)(10,15)(14,16)(17,20,27,29,22,28,21,19)(18,24,31,23,26,32,25,30)

Polycyclic presentation:

2^9|a^2=h,b^2=eghi,c^2=e,d^2=fh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dhi,d^c=dh,g^c=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, 233250);