#383126

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290322);