#391991

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

717 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:

(5,6,10,17,11,15,27,16)(7,12,8,20,19,13,22,29)(9,14,25,33,26,32,24,34)(18,28,21,31,35,30,23,36);(1,2,3,4)(5,7,11,19)(6,12)(8,10,22,27)(9,21)(13,15)(14,31,32,36)(16,20)(17,29)(18,25)(23,26)(24,35)(28,34,30,33);(5,8,10,19,11,22,27,7)(6,13,16,20,15,12,17,29)(9,23,24,35,26,21,25,18)(14,30,33,36,32,28,34,31);(5,9)(6,14)(7,18)(8,21)(10,25)(11,26)(12,28)(13,30)(15,32)(16,34)(17,33)(19,35)(20,31)(22,23)(24,27)(29,36)

Polycyclic presentation:

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

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, 299187);