#499111

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×#532	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1143 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)(5,6,12,16)(7,13,10,25)(8,19,22,28)(9,23,11,15)(14,26,34,17)(18,30,24,29)(20,32,35,33)(21,31,27,36);(1,2)(3,4)(5,7)(6,13)(8,20)(9,18)(10,12)(11,24)(14,31)(15,29)(16,25)(17,21)(19,32)(22,35)(23,30)(26,27)(28,33)(34,36);(1,3,4,2)(5,8)(6,14)(7,17)(9,21)(10,26)(11,27)(12,22)(13,28)(15,32)(16,34)(18,35)(19,25)(20,24)(23,33)(29,31)(30,36);(5,9,12,11)(6,15,16,23)(7,18,10,24)(8,21,22,27)(13,29,25,30)(14,32,34,33)(17,35,26,20)(19,36,28,31)

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,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, 406307);