#357858

order	512
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2610 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303832, #72303843, #72303853
maximal normal subgroup of	#72303832, #72303843, #72303853
maximal quotient of	#72303942

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265054);