#418467

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1723 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256053, #72262403, #72262421
maximal normal subgroup of	#72256053, #72262403, #72262421

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325663);