#299809

order	512
nilpotency class	3
isoclinism	isoclinic class of #299805
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	#19073	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

943 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)(4,15)(5,14)(6,12)(7,11)(8,13)(10,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,31)(30,32);(1,3,6,13,7,4,16,14)(2,8,10,9,11,5,12,15)(19,26)(20,29)(23,28)(25,31);(1,4)(2,9)(3,7)(5,10)(6,14)(8,12)(11,15)(13,16)(17,19,20,27,21,26,29,32)(18,22,23,25,24,30,28,31);(1,5,6,15,7,8,16,9)(2,4,10,14,11,3,12,13)(19,26)(22,30)(25,31)(27,32)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,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,d^c=di,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207005);