#299740

order	512
nilpotency class	3
isoclinism	isoclinic class of #299717
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	#10104	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

882 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,g^c=ghi

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, 206936);