#248316

order	512
nilpotency class	3
isoclinism	isoclinic class of #248263
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	#36051	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1100 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)(4,13)(5,9)(6,14)(7,10)(11,15)(12,16)(17,18,23,28)(19,29,20,31)(21,27,22,26)(24,32,25,30);(1,3)(2,8)(4,6)(5,11)(7,12)(9,15)(10,16)(13,14)(17,19,23,20)(18,24,28,25)(21,32,22,30)(26,31,27,29);(1,4)(2,8)(3,5)(6,12)(7,11)(9,15)(10,16)(13,14)(17,19,23,20)(18,24,28,25)(21,30,22,32)(26,29,27,31);(1,5,7,6)(2,9,10,14)(3,11,12,4)(8,15,16,13)(17,20,23,19)(18,25,28,24)(21,30,22,32)(26,29,27,31)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,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=ghi,c^b=cg,e^b=eh,f^b=fhi,f^c=fi,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155512);