#418325

order	512
nilpotency class	3
isoclinism	isoclinic class of #417579
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
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1781 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)(7,11)(8,18)(9,21)(10,13)(12,20)(14,22)(15,29)(16,24)(17,33)(19,28)(23,27)(25,30)(26,32)(31,36)(34,35);(1,2)(3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,25)(13,24)(14,26)(18,29)(19,31)(20,30)(21,33)(22,32)(23,34)(27,35)(28,36);(9,22)(11,24)(14,21)(17,32)(18,28)(20,23)(25,35)(30,34);(1,3,4,2)(5,8,10,19)(6,12,13,27)(7,15,16,31)(9,23,22,20)(11,25,24,35)(14,28,21,18)(17,34,32,30)(26,36,33,29)

Polycyclic presentation:

2^9|d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,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, 325521);