#301989

order	512
nilpotency class	3
isoclinism	isoclinic class of #301973
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	C₂³×C₄	C₂⁴
prime radical	#62280	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1024 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)(4,7)(5,8)(6,11)(10,14)(12,16)(13,15)(17,18)(19,27)(20,21)(22,25)(23,26)(24,30)(28,32)(29,31);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,14,15,16)(17,19,21,30,22,31,23,28)(18,24,26,27,25,32,20,29);(1,4)(2,7)(3,10,11,12)(5,13)(6,14,9,16)(8,15)(18,25)(19,28,31,30)(20,26)(24,27,32,29);(2,8)(6,9)(7,15)(14,16)(17,20,23,25,22,26,21,18)(19,29,28,32,31,27,30,24)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209185);