#478477

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

797 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)(5,6,11,16)(7,17)(8,22,25,33)(9,26,24,13)(10,14,28,15)(12,20)(18,35)(19,32)(21,31)(23,30)(27,29)(34,36);(5,7,11,21)(6,12)(8,23,25,36)(9,27,24,18)(10,19,28,20)(13,30)(14,17)(15,31)(16,32)(22,29)(26,34)(33,35);(5,8,11,25)(6,13,16,26)(7,18,21,27)(9,28,24,10)(12,29,32,35)(14,22,15,33)(17,34,31,30)(19,23,20,36);(2,4)(5,9,10,8,11,24,28,25)(6,13,14,33,16,26,15,22)(7,18,19,36,21,27,20,23)(12,30,17,35,32,34,31,29)

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=i,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=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, 385673);