#419030

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2313 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)(7,15)(8,19,22,12)(9,23)(10,14)(11,17)(13,25)(16,28,33,31)(18,32)(20,26,34,36)(21,29,24,30)(27,35);(1,2)(3,4)(5,7,9,17,10,18,25,35)(6,11,13,15,14,27,23,32)(8,20,21,33,22,34,24,16)(12,28,29,36,19,31,30,26);(2,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,17)(16,26)(18,27)(19,22)(20,28)(21,29)(23,25)(24,30)(31,34)(32,35)(33,36);(5,8)(6,12)(7,16)(9,24)(10,22)(11,26)(13,30)(14,19)(15,31)(17,34)(18,33)(20,35)(21,25)(23,29)(27,36)(28,32)

Polycyclic presentation:

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

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, 326226);