#248146

order	512
nilpotency class	3
isoclinism	isoclinic class of #248071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61890	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 155342);