#402022

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

771 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)(5,6,11,16,12,18,10,17)(7,15,20,30,21,29,9,13)(8,22,26,14,27,32,25,33)(19,28,23,31,34,36,24,35);(5,7,10,9,12,21,11,20)(6,13,16,15,18,30,17,29)(8,23,26,34,27,24,25,19)(14,31,22,28,33,35,32,36);(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,27)(13,28)(15,31)(16,32)(17,22)(18,33)(20,23)(21,34)(29,35)(30,36);(2,4)(5,9,10,21,12,20,11,7)(6,15,16,30,18,29,17,13)(8,24,25,34,27,23,26,19)(14,31,32,36,33,35,22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309218);