#417707

order	512
nilpotency class	3
isoclinism	isoclinic class of #417704
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2337 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^c=fi,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, 324903);