#420051

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	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1364 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,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, 327247);