#300427

order	512
nilpotency class	3
isoclinism	isoclinic class of #300397
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	#62557	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

859 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)(3,11)(4,13)(5,9)(6,15)(7,10)(8,12)(14,16)(17,18)(19,31)(20,37)(21,26)(22,29)(23,28)(24,30)(25,34)(27,40)(32,41)(33,47)(35,43)(36,42)(38,48)(39,45)(44,46);(1,3,5,12)(2,8,9,11)(4,7,14,6)(10,13,15,16)(17,19,22,34,24,36,23,35)(18,25,28,31,30,43,29,42)(20,38,21,33,40,46,39,32)(26,44,37,47,45,48,27,41);(1,4,6,12,5,14,7,3)(2,8,10,13,9,11,15,16)(17,20,23,39,24,40,22,21)(18,26,29,27,30,45,28,37)(19,32,34,38,36,33,35,46)(25,41,31,44,43,47,42,48);(2,9)(8,11)(10,15)(13,16)(17,21,22,40,24,39,23,20)(18,27,28,26,30,37,29,45)(19,33,34,46,36,32,35,38)(25,41,31,44,43,47,42,48)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,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,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207623);