#388087

order	512
nilpotency class	3
isoclinism	isoclinic class of #31418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1069 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)(5,6,10,15,11,12,9,22)(7,13,16,26,18,25,24,28)(8,19,21,29,20,14,23,30)(17,32,31,36,33,27,34,35);(6,12)(7,16)(8,20)(9,10)(13,26)(14,19)(17,31)(18,24)(25,28)(27,36)(32,35)(33,34);(5,7,9,16,11,18,10,24)(6,13,15,28,12,25,22,26)(8,17,23,31,20,33,21,34)(14,27,30,36,19,32,29,35);(5,8,9,23,11,20,10,21)(6,14,15,30,12,19,22,29)(7,17,16,31,18,33,24,34)(13,27,28,36,25,32,26,35)

Polycyclic presentation:

2^9|a^2=h,c^2=efg,d^2=efg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,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, 295283);