#422716

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

1044 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:

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

Polycyclic presentation:

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

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, 329912);