#357861

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	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2530 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=i,h^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	#72303821, #72303826, #72303837
maximal normal subgroup of	#72303821, #72303826, #72303837
maximal quotient of	#72303959

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265057);