#418908

order	512
nilpotency class	3
isoclinism	isoclinic class of #417690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2215 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228037, #72228038
maximal normal subgroup of	#72228037, #72228038

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326104);