#247923

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	C₂³	C₂×#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1080 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)(19,23)(20,28)(21,30)(22,26)(24,29)(25,31)(27,32);(1,3,6,4)(2,7,10,8)(5,14,16,12)(9,13,15,11)(17,19,22,20)(18,23,26,28)(21,29,31,27)(24,25,32,30);(17,20,22,19)(18,24,26,32)(21,27,31,29)(23,30,28,25);(1,4,6,3)(2,8,10,7)(5,12,16,14)(9,11,15,13)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=h,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, 155119);