#274069

order	512
nilpotency class	3
isoclinism	isoclinic class of #273947
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	#24670	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

848 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=i,d^2=i,e^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fhi,g^c=gh,e^d=ei,f^d=fhi

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, 181265);