#427981

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1266 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 335177);