#299117

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
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

953 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)(10,14)(12,17)(13,16)(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)(6,11,19,12,14,20,13,15)(21,23,26,24,27,35,25,34)(22,28,31,29,32,36,30,33);(2,4)(7,17)(8,18)(11,20)(12,15)(21,24,25,23,27,34,26,35)(22,29,30,28,32,33,31,36);(5,8,9,17,10,18,16,7)(6,12,13,11,14,15,19,20)(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=fi,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,d^c=dh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206313);