#296805

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

765 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)(9,10,13,16)(11,12,14,15)(17,18,21,25)(19,24,27,23)(20,29,28,32)(22,31,30,26);(2,5)(6,7)(9,11,13,14)(10,15,16,12)(17,19,21,27)(18,23,25,24)(20,30,28,22)(26,29,31,32);(9,12,14,16,13,15,11,10)(17,20,22,19,21,28,30,27)(18,24,26,29,25,23,31,32);(1,3,4,8)(2,6,5,7)(10,16)(12,15)(18,25)(23,24)(26,31)(29,32)

Polycyclic presentation:

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

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, 204001);