#416319

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

720 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)(5,6,11,17,12,18,10,16)(7,14,8,21,20,29,23,13)(9,24,26,32,27,31,25,15)(19,28,22,30,33,34,35,36);(5,7,12,20)(6,13)(8,11,23,10)(9,19,27,33)(14,16)(15,28)(17,29)(18,21)(22,26,35,25)(24,36)(30,31)(32,34);(2,4)(5,8,10,20,12,23,11,7)(6,14,16,21,18,29,17,13)(9,22,25,33,27,35,26,19)(15,30,31,34,32,36,24,28);(5,9,10,25,12,27,11,26)(6,15,17,24,18,32,16,31)(7,19,8,22,20,33,23,35)(13,28,29,36,21,34,14,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323515);