#406017

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,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, 313213);