#462286

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

659 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:

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

Polycyclic presentation:

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

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, 369482);