#300522

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	#3558	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

854 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=ehi,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, 207718);