#285532

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	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

629 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,d^2=h,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, 192728);