#422132

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1009 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=e,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=de,e^b=ei,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, 329328);