#245124

order	512
nilpotency class	3
isoclinism	isoclinic class of #245105
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	#18296	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|b^2=hi,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=di,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, 152320);