#299522

order	512
nilpotency class	3
isoclinism	isoclinic class of #299505
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	#60820	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

946 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)(3,5)(4,11)(6,10)(7,13)(8,14)(9,15)(12,16)(17,18,24,19)(20,21,31,32)(22,30,23,27)(25,29,28,26);(1,3,5,2)(4,9,12,7)(6,10,14,8)(11,13,16,15)(17,19,22,30,24,18,23,27)(20,21,26,28,31,32,29,25);(1,4,6,13,5,12,14,15)(2,7,8,11,3,9,10,16)(17,20,22,26,24,31,23,29)(18,25,27,21,19,28,30,32);(2,3)(4,12)(8,10)(13,15)(17,21,23,25,24,32,22,28)(18,26,30,20,19,29,27,31)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=ei,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,d^b=di,g^b=gh,d^c=dhi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206718);