#319036

order	512
nilpotency class	3
isoclinism	isoclinic class of #298603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25720	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

945 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eghi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226232);