#239521

order	512
nilpotency class	3
isoclinism	isoclinic class of #232393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1255 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=di,f^b=fi,d^c=di,f^c=fh

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, 146717);