#6369817

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

5198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=i,b^a=bf,h^a=hi,f^b=fi,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280474, #72280582, #72280597, …
maximal normal subgroup of	#72280474, #72280582, #72280597, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277013);