#499114

order	512
nilpotency class	4
isoclinism	isoclinic class of #498904
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×Q₈	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1175 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)(5,6,11,16)(7,17,20,30)(8,15,10,13)(9,22,24,14)(12,23,28,19)(18,33,25,34)(21,29,26,32)(27,35,31,36);(1,2)(3,4)(7,18)(8,10)(12,27)(13,15)(17,33)(19,36)(20,25)(21,26)(23,35)(28,31)(29,32)(30,34);(1,3,4,2)(5,7)(6,12)(8,18)(9,23)(10,25)(11,20)(13,27)(14,30)(15,31)(16,28)(17,22)(19,24)(21,36)(26,35)(29,33)(32,34);(5,8,11,10)(6,13,16,15)(7,18,20,25)(9,21,24,26)(12,27,28,31)(14,29,22,32)(17,34,30,33)(19,35,23,36)

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,g^c=gi,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, 406310);