#118628

order	512
nilpotency class	3
isoclinism	isoclinic class of #10208
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(2,7)(4,5)(8,10)(9,13)(11,14)(15,16)(17,18,22,26,23,27,21,36)(19,24,31,39,30,40,29,45)(20,32,35,25,34,42,37,41)(28,43,44,38,33,47,46,48);(1,2,4,7,6,9,5,13)(3,10,11,8,12,15,14,16)(17,19)(18,24)(20,33,34,28)(21,29)(22,31)(23,30)(25,38,41,48)(26,39)(27,40)(32,43,42,47)(35,46,37,44)(36,45);(1,3)(2,8)(4,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,20)(18,25)(19,28)(21,37)(22,35)(23,34)(24,38)(26,42)(27,41)(29,46)(30,33)(31,44)(32,36)(39,47)(40,48)(43,45)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=ei,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25824);