#271991

order	512
nilpotency class	3
isoclinism	isoclinic class of #271963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#537	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

791 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:

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

Polycyclic presentation:

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

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, 179187);