#391208

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂×#2018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1109 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:

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

Polycyclic presentation:

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

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, 298404);