#379516

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

898 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286712);