#423782

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1289 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^c=gi,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, 330978);