#283608

order	512
nilpotency class	3
isoclinism	isoclinic class of #283579
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

616 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 190804);