#245194

order	512
nilpotency class	3
isoclinism	isoclinic class of #245168
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3192	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^c=fi,f^d=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, 152390);