#299284

order	512
nilpotency class	3
isoclinism	isoclinic class of #299253
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60714	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

965 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206480);