#499102

order	512
nilpotency class	4
isoclinism	isoclinic class of #498890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406298);