#419811

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1554 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 327007);