#232512

order	512
nilpotency class	3
isoclinism	isoclinic class of #9980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#392
prime radical	#36652	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1390 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,45)(31,39)(33,40)(34,48)(35,42)(36,44)(37,43)(38,46)(41,47);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19)(18,24)(20,34)(21,31)(22,32)(23,33)(25,41)(26,29)(27,40)(28,39)(30,35)(36,46)(37,47)(38,43)(42,48)(44,45);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,42)(27,43)(28,44)(29,41)(31,46)(32,34)(33,47)(39,48)(40,45);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,29,39,40)(25,42,44,43)(30,46,47,34)(38,41,48,45)

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139708);