#247971

order	512
nilpotency class	3
isoclinism	isoclinic class of #247879
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1067 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=hi,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,e^b=eh,f^b=fi,f^c=fi,e^d=eh

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, 155167);