#493699

order	512
nilpotency class	4
isoclinism	isoclinic class of #480268
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1248 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)(7,11)(8,18)(9,21,22,13)(10,14)(12,19,26,33)(15,27)(16,24)(17,31,30,25)(20,32)(23,29,34,36)(28,35);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,23)(13,25)(14,24)(18,27)(19,29)(20,28)(21,31)(22,30)(26,34)(32,35)(33,36);(5,8,10,20)(6,12,14,26)(7,15,16,28)(9,19,22,33)(11,23,24,34)(13,18,21,32)(17,29,30,36)(25,27,31,35);(7,16)(11,24)(15,28)(17,30)(23,34)(25,31)(27,35)(29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,g^c=gh,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, 400895);