#416439

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

960 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=e,d^2=hi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,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, 323635);