#392687

order	512
nilpotency class	3
isoclinism	isoclinic class of #196092
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3809 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)(5,6,10,15,11,16,9,14)(7,17,20,28,21,27,19,12)(8,22,23,13,25,29,24,30)(18,26,32,35,34,36,33,31);(5,7)(6,12,16,28)(8,18)(9,19)(10,20)(11,21)(13,26,30,36)(14,27,15,17)(22,31,29,35)(23,32)(24,33)(25,34);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,29)(15,22)(16,30)(17,31)(19,32)(20,33)(21,34)(27,35)(28,36);(2,4)(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,25,24)(12,27,28,17)(13,29,30,22)(18,32,34,33)(26,35,36,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260846, #72306319, #72306505
maximal normal subgroup of	#72260846, #72306319, #72306505
maximal quotient of	#72260864

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299883);