#452544

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

895 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)(5,6,12,15,10,16,11,9)(7,17,20,19,21,13,28,27)(8,22,24,14,26,32,25,31)(18,30,23,29,34,36,35,33);(5,7,11,20,10,21,12,28)(6,13,15,19,16,17,9,27)(8,23,24,18,26,35,25,34)(14,30,22,29,31,36,32,33);(5,8,12,24,10,26,11,25)(6,14,15,32,16,31,9,22)(7,18,20,23,21,34,28,35)(13,29,27,36,17,33,19,30);(2,4)(5,9,11,16,10,15,12,6)(7,19,28,17,21,27,20,13)(8,22,25,31,26,32,24,14)(18,30,35,33,34,36,23,29)

Polycyclic presentation:

2^9|a^2=egh,b^2=f,c^2=ef,d^2=efi,f^2=i,g^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,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

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, 359740);