#414033

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1291 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=gh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,g^c=gi,h^c=hi,f^d=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, 321229);