#419820

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1683 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)(7,15,17,11)(8,19,20,34)(9,14)(10,13)(12,21,27,35)(16,30)(18,29,22,25)(23,28)(24,31)(26,36)(32,33);(1,2)(3,4)(5,7)(6,11)(8,16)(9,22)(10,17)(12,26)(13,15)(14,29)(18,23)(19,33)(20,32)(21,36)(24,27)(25,28)(30,34)(31,35);(2,4)(9,23)(11,15)(14,28)(18,22)(19,34)(21,35)(24,26)(31,36);(5,8,10,20)(6,12)(7,16,17,32)(9,21,23,35)(11,24)(13,27)(14,19)(15,26)(18,31,22,36)(25,33)(28,34)(29,30)

Polycyclic presentation:

2^9|a^2=f,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,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, 327016);