#347225

order	512
nilpotency class	3
isoclinism	isoclinic class of #346934
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	#63031	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2069 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=hi,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,e^c=ei,f^c=fi,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, 254421);