#477323

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

882 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384519);