#469775

order	512
nilpotency class	3
isoclinism	isoclinic class of #464719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1079 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376971);