#392700

order	512
nilpotency class	3
isoclinism	isoclinic class of #196320
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35088	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3905 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,f^b=fi,h^b=hi,e^c=ei,h^c=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, 299896);