#285549

order	512
nilpotency class	3
isoclinism	isoclinic class of #285499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

583 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 192745);