#418308

order	512
nilpotency class	3
isoclinism	isoclinic class of #417601
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1731 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)(7,12)(8,20)(9,14)(10,23)(11,15)(13,22)(16,25)(17,32)(18,27)(19,28)(21,30)(24,29)(26,31)(33,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,17)(9,16)(10,18)(11,19)(13,26)(14,25)(15,28)(20,32)(21,33)(22,31)(23,27)(24,34)(29,36)(30,35);(7,16)(9,10)(12,25)(14,23)(17,31)(18,19)(20,30)(22,24)(26,35)(27,28)(32,36)(33,34);(1,3,4,2)(5,8)(6,13)(7,17)(9,22)(10,24)(11,21)(12,26)(14,30)(15,29)(16,31)(18,34)(19,33)(20,23)(25,35)(27,32)(28,36)

Polycyclic presentation:

2^9|d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,f^d=fi,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, 325504);