#147120

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=di,e^b=ei,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224874, #72224942, #72224952, …
maximal normal subgroup of	#72224874, #72224942, #72224952, …
maximal quotient of	#72224869, #72224872

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54316);