#268087

order	512
nilpotency class	3
isoclinism	isoclinic class of #268043
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	C₂×#496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

825 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:

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

Polycyclic presentation:

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

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, 175283);