#347384

order	512
nilpotency class	3
isoclinism	isoclinic class of #347322
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#537	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1956 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254580);