#423189

order	512
nilpotency class	3
isoclinism	isoclinic class of #419843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1315 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:

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

Polycyclic presentation:

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

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, 330385);