#478428

order	512
nilpotency class	3
isoclinism	isoclinic class of #476609
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

956 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:

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

Polycyclic presentation:

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

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, 385624);