#365136

order	512
nilpotency class	3
isoclinism	isoclinic class of #24420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1862 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:

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

Polycyclic presentation:

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

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, 272332);