#287334

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ 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:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=f,c^2=hi,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 194530);