#477833

order	512
nilpotency class	3
isoclinism	isoclinic class of #476293
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

708 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:

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

Polycyclic presentation:

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

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, 385029);