#381893

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1214 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)(5,6,10,15,12,16,11,9)(7,17,19,28,20,31,26,13)(8,21,22,14,24,25,23,30)(18,27,33,36,34,29,35,32);(5,7)(6,13)(8,18)(9,17)(10,26)(11,19)(12,20)(14,29)(15,31)(16,28)(21,32)(22,35)(23,33)(24,34)(25,36)(27,30);(2,4)(5,8,12,24)(6,14,16,30)(7,18,20,34)(9,25,15,21)(10,23,11,22)(13,27,28,29)(17,32,31,36)(19,35,26,33);(5,9,11,16,12,15,10,6)(7,17,19,28,20,31,26,13)(8,14,23,21,24,30,22,25)(18,29,33,32,34,27,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289089);