#348031

order	512
nilpotency class	3
isoclinism	isoclinic class of #348018
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18337	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

802 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:

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

Polycyclic presentation:

2^9|b^2=ei,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,f^c=fi,g^c=ghi,f^d=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255227);