#500197

order	512
nilpotency class	4
isoclinism	isoclinic class of #499867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#46310	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1051 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,17)(13,25)(14,28)(15,19)(16,22)(18,29)(21,37)(23,32)(24,36)(26,33)(27,34)(30,40)(31,38)(35,39);(2,5)(6,8)(11,21)(14,29)(16,31)(18,28)(20,37)(22,38)(23,30)(24,27)(26,39)(32,40)(33,35)(34,36);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,26)(14,30)(15,23)(17,32)(18,35)(19,33)(20,25)(21,29)(24,39)(27,38)(28,31)(34,37)(36,40);(2,5)(6,8)(9,12,15,13)(10,17,19,25)(11,22,23,26)(14,24,29,27)(16,32,33,20)(18,34,28,36)(21,38,30,39)(31,40,35,37)

Polycyclic presentation:

2^9|c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=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, 407393);