#285113

order	512
nilpotency class	3
isoclinism	isoclinic class of #285051
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

647 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=hi,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=di,e^b=eh,g^b=gh,f^c=fi,g^c=ghi,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192309);