#451177

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

755 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)(5,6,12,7,10,15,11,18)(8,16,26,28,20,31,21,13)(9,22,25,17,23,14,24,29)(19,30,34,27,32,36,33,35);(5,7,11,15,10,18,12,6)(8,16,21,28,20,31,26,13)(9,17,24,14,23,29,25,22)(19,30,33,27,32,36,34,35);(5,8,10,20)(6,13)(7,16)(9,19,23,32)(11,21,12,26)(14,27)(15,28)(17,30)(18,31)(22,35)(24,33,25,34)(29,36);(2,4)(5,9,10,23)(6,14,15,22)(7,17,18,29)(8,19,20,32)(11,25,12,24)(13,27,28,35)(16,30,31,36)(21,34,26,33)

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=e,d^2=i,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=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358373);