#493714

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1210 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400910);