#288639

order	512
nilpotency class	3
isoclinism	isoclinic class of #288611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

596 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:

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

Polycyclic presentation:

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

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, 195835);