#465851

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	#23438	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1035 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373047);