#478305

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

774 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)(5,6,11,16,12,17,10,33)(7,18,22,29,23,34,21,13)(8,24,26,20,27,36,25,14)(9,28,30,19,32,15,31,35);(5,7,10,21,12,23,11,22)(6,13,16,18,17,29,33,34)(8,15,25,19,27,28,26,35)(9,14,30,24,32,20,31,36);(5,8,12,27)(6,14,17,20)(7,19,23,35)(9,29,32,13)(10,26,11,25)(15,22,28,21)(16,36,33,24)(18,31,34,30);(2,4)(5,9)(6,15)(7,20)(8,18)(10,31)(11,30)(12,32)(13,25)(14,23)(16,35)(17,28)(19,33)(21,24)(22,36)(26,29)(27,34)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,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, 385501);