#347195

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

2067 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,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, 254391);