#412958

order	512
nilpotency class	3
isoclinism	isoclinic class of #404632
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

978 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi

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, 320154);