#298993

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

959 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,15)(8,11)(9,19)(10,12)(13,17)(14,16)(18,20)(21,22,27,32)(23,33,35,29)(24,36,34,28)(25,31,26,30);(1,2)(3,4)(5,7,16,18,10,17,9,8)(6,11,19,13,12,20,14,15)(21,23,26,24,27,35,25,34)(22,28,31,29,32,36,30,33);(2,4)(6,12)(11,20)(13,15)(14,19)(21,24,26,35,27,34,25,23)(22,29,31,36,32,33,30,28);(5,8,9,17,10,18,16,7)(6,13,14,11,12,15,19,20)(21,23,25,34,27,35,26,24)(22,28,30,33,32,36,31,29)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fhi,d^2=e,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^b=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, 206189);