#235512

order	512
nilpotency class	3
isoclinism	isoclinic class of #235504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60962	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1230 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)(3,11)(4,7)(5,9)(6,10)(8,12)(13,15)(14,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,44)(30,38)(32,41)(33,39)(34,42)(35,40)(37,46)(43,47)(45,48);(2,7)(4,5)(8,11)(9,10)(12,14)(15,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,37)(26,41)(27,38)(28,36)(33,46)(34,45)(35,47)(39,43)(40,48)(42,44);(1,3,6,13)(2,8,10,16)(4,12,5,14)(7,15,9,11)(17,20,23,35)(18,25,27,40)(19,29,32,47)(21,34,22,33)(24,37,38,48)(26,42,36,39)(28,43,41,44)(30,46,31,45);(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,35,34)(24,28,38,41)(25,39,40,42)(29,45,47,46)(37,44,48,43)

Polycyclic presentation:

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

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, 142708);