#315719

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

688 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=e,c^2=f,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222915);