#300370

order	512
nilpotency class	3
isoclinism	isoclinic class of #300333
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	#3498	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1029 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,g^b=gi,d^c=dhi,g^c=gi

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, 207566);