#300296

order	512
nilpotency class	3
isoclinism	isoclinic class of #300269
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

867 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=f,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,g^b=gi,d^c=di,g^c=gi

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, 207492);