#300527

order	512
nilpotency class	3
isoclinism	isoclinic class of #300501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3561	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

858 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:

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

Polycyclic presentation:

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

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, 207723);