#299389

order	512
nilpotency class	3
isoclinism	isoclinic class of #299341
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	#3498	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

968 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18)(19,20)(21,27)(22,26)(23,28)(24,25)(29,31)(30,32);(2,6)(7,14)(8,13)(9,15)(17,18,22,27,23,28,21,26)(19,20,25,31,30,32,29,24);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,19,22,25,23,30,21,29)(18,24,27,20,28,31,26,32);(2,6)(3,10)(8,13)(11,16)(17,20,22,31,23,32,21,24)(18,25,27,30,28,29,26,19)

Polycyclic presentation:

2^9|b^2=ei,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,g^b=gh,d^c=di,g^c=gh

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, 206585);