#352308

order	512
nilpotency class	3
isoclinism	isoclinic class of #352300
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2856
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

775 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)(3,6)(7,8)(9,10,15,20)(11,21,22,32)(12,26,28,34)(13,19,14,18)(16,25,30,24)(17,29,33,27)(23,37,36,31)(35,39,40,38);(9,11,13,24,15,22,14,25)(10,16,18,21,20,30,19,32)(12,23,27,38,28,36,29,39)(17,31,26,35,33,37,34,40);(1,2,6,8,4,7,3,5)(11,22)(16,30)(21,32)(23,36)(24,25)(31,37)(35,40)(38,39);(9,12)(10,17)(11,23)(13,27)(14,29)(15,28)(16,31)(18,26)(19,34)(20,33)(21,35)(22,36)(24,38)(25,39)(30,37)(32,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,g^d=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259504);