#299197

order	512
nilpotency class	3
isoclinism	isoclinic class of #299195
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	#2619	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=e,e^2=h,f^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^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206393);