#283831

order	512
nilpotency class	3
isoclinism	isoclinic class of #283771
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	#3035	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=hi,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,d^c=di,f^c=fh,g^c=ghi,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, 191027);