#512302

order	512
nilpotency class	4
isoclinism	isoclinic class of #512072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

605 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)(7,15)(8,18)(9,22)(12,26)(13,28)(14,25)(16,32)(17,21)(19,34)(20,35)(23,33)(24,31)(27,29)(30,36);(1,2,4,3)(5,6)(7,12)(8,19)(9,14)(10,11)(13,20)(15,31)(16,24)(17,27)(18,34)(21,36)(22,25)(23,30)(26,32)(28,35)(29,33);(6,11)(8,20)(12,24)(14,25)(15,32)(18,35)(21,33)(27,30);(1,3,4,2)(5,7,9,23,10,16,22,17)(6,12,14,30,11,24,25,27)(8,21,18,32,20,33,35,15)(13,29,28,31,19,36,34,26)

Polycyclic presentation:

2^9|b^2=ef,d^2=eh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=ghi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419498);