#412957

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×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

946 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)(7,16)(8,20)(9,10)(13,17)(14,29)(18,33)(19,24)(21,22)(25,28)(26,30)(27,35);(1,2)(3,4)(5,6,9,23,11,12,10,15)(7,17,16,25,19,28,24,13)(8,14,21,29,22,26,20,30)(18,32,31,35,33,36,34,27);(2,4)(5,7)(6,13,12,25)(8,18)(9,24)(10,16)(11,19)(14,27,26,35)(15,17,23,28)(20,31)(21,34)(22,33)(29,36,30,32);(5,8,10,20,11,22,9,21)(6,14,15,30,12,26,23,29)(7,18,16,31,19,33,24,34)(13,27,17,32,25,35,28,36)

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=h,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, 320153);