#428065

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1157 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)(5,6,11,16)(7,17,20,30)(8,21,24,28)(9,15,10,14)(12,25,27,18)(13,22,29,23)(19,33,35,26)(31,34,36,32);(7,18,20,25)(12,17,27,30)(19,34,35,32)(26,36,33,31);(5,7)(6,12)(8,19)(9,25)(10,18)(11,20)(13,26)(14,30)(15,17)(16,27)(21,31)(22,34)(23,32)(24,35)(28,36)(29,33);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,28,16,29,15,21)(7,19,25,34,20,35,18,32)(12,26,30,36,27,33,17,31)

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=ef,f^2=i,g^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,e^c=ei,f^c=fi,g^c=gi,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, 335261);