#147984

order	512
nilpotency class	3
isoclinism	isoclinic class of #11115
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1924
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fi,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55180);