#300345

order	512
nilpotency class	3
isoclinism	isoclinic class of #300333
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1089 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)(4,13)(5,7)(6,12)(8,11)(10,15)(14,16)(17,18)(19,26)(20,30)(21,24)(22,25)(23,29)(27,28)(31,32);(1,3,4,12,5,10,11,16)(2,6,8,9,7,14,13,15)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,28)(31,32);(2,7)(3,10)(8,13)(12,16)(17,19,20,29,22,28,21,31)(18,23,24,26,25,32,30,27);(1,3,11,16,5,10,4,12)(2,6,13,15,7,14,8,9)(19,28)(23,32)(26,27)(29,31)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,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=dhi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207541);