#391598

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

839 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)(5,6,11,13,9,16,10,15)(7,12,20,27,19,28,17,26)(8,14,22,29,24,32,23,31)(18,25,21,35,34,36,33,30);(5,7)(6,12)(8,21)(9,19)(10,20)(11,17)(13,26)(14,30)(15,27)(16,28)(18,22)(23,34)(24,33)(25,31)(29,36)(32,35);(6,13)(7,17)(10,11)(14,31)(15,16)(18,33)(19,20)(21,34)(22,23)(25,36)(26,27)(29,32);(5,8,10,23,9,24,11,22)(6,14,15,31,16,32,13,29)(7,18,17,33,19,34,20,21)(12,25,26,30,28,36,27,35)

Polycyclic presentation:

2^9|a^2=h,d^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 298794);