#346857

order	512
nilpotency class	3
isoclinism	isoclinic class of #346786
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	#62965	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2064 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)(8,15)(9,12)(10,13)(14,16)(17,18,21,25)(19,26,28,31)(20,24,22,29)(23,30,32,27);(1,2)(3,7)(4,8)(5,12)(6,9)(10,14)(11,16)(13,15)(17,18)(19,23)(20,24)(21,25)(22,29)(26,30)(27,31)(28,32);(1,3,6,5)(2,7,9,12)(4,10,11,13)(8,14,16,15)(17,19,21,28)(18,23,25,32)(20,27,22,30)(24,31,29,26);(1,4,6,11)(2,8,9,16)(3,10,5,13)(7,14,12,15)(17,20,21,22)(18,24,25,29)(19,27,28,30)(23,31,32,26)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254053);