#271937

order	512
nilpotency class	3
isoclinism	isoclinic class of #271899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60597	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

921 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179133);