#388730

order	512
nilpotency class	3
isoclinism	isoclinic class of #30437
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

887 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)(7,17)(8,22)(9,13)(10,15)(11,16)(12,20)(14,24)(18,31)(19,34)(21,28)(23,36)(25,32)(26,30)(27,33)(29,35);(5,7,9,18,11,20,10,21)(6,12,15,31,16,17,13,28)(8,23,24,33,25,19,26,35)(14,29,22,34,30,27,32,36);(1,3,4,2)(6,13)(7,18)(9,10)(14,22)(15,16)(19,33)(20,21)(23,35)(24,26)(28,31)(30,32)(34,36);(5,8)(6,14)(7,19)(9,26)(10,24)(11,25)(12,27)(13,22)(15,32)(16,30)(17,29)(18,33)(20,23)(21,35)(28,36)(31,34)

Polycyclic presentation:

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

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, 295926);