#479251

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

639 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)(5,6,12,16)(7,17,21,9)(8,22)(10,27,11,15)(13,20,29,19)(14,24)(18,23)(25,32)(26,36)(28,31)(30,35)(33,34);(5,7,12,21)(6,13,16,29)(8,18,26,35)(9,27,17,15)(10,20,11,19)(14,23,32,30)(22,33,36,28)(24,31,25,34);(5,8,10,24,12,26,11,25)(6,14,15,22,16,32,27,36)(7,18,19,34,21,35,20,31)(9,28,13,30,17,33,29,23);(2,4)(5,9,11,13,12,17,10,29)(6,7,15,20,16,21,27,19)(8,23,25,28,26,30,24,33)(14,31,22,35,32,34,36,18)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386447);