#348025

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	#537	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

776 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,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, 255221);