#462136

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

681 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)(5,6,12,17,11,18,10,16)(7,19,21,28,23,32,22,13)(8,20,25,34,27,33,26,14)(9,15,31,24,30,36,29,35);(5,7,10,21,11,23,12,22)(6,13,17,32,18,28,16,19)(8,24,26,36,27,35,25,15)(9,20,29,34,30,33,31,14);(5,8,10,26,11,27,12,25)(6,14,17,20,18,34,16,33)(7,15,21,24,23,36,22,35)(9,28,29,19,30,13,31,32);(2,4)(5,9,10,29,11,30,12,31)(6,15,16,35,18,36,17,24)(7,20,21,34,23,33,22,14)(8,19,25,28,27,32,26,13)

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=f,d^2=e,e^2=i,f^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,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, 369332);