#383142

order	512
nilpotency class	3
isoclinism	isoclinic class of #213752
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1978 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:

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

Polycyclic presentation:

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

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, 290338);