#392709

order	512
nilpotency class	3
isoclinism	isoclinic class of #32950
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

3623 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)(3,4)(5,6)(7,17)(8,23,26,28)(9,25,27,24)(10,15)(11,14)(12,16)(13,20)(18,33,35,19)(21,30)(22,29)(31,34,36,32);(5,7,10,20,12,22,11,21)(6,13,14,17,16,30,15,29)(8,18,25,32,26,35,24,34)(9,19,28,31,27,33,23,36);(2,4)(5,8)(6,9)(7,18)(10,25)(11,24)(12,26)(13,19)(14,28)(15,23)(16,27)(17,31)(20,32)(21,34)(22,35)(29,36)(30,33);(5,9)(6,8)(7,19)(10,28)(11,23)(12,27)(13,18)(14,25)(15,24)(16,26)(17,32)(20,31)(21,36)(22,33)(29,34)(30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299905);