#147968

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
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×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

690 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)(5,6,9,15,10,14,24,30,11,13,25,29,8,20,22,33)(7,16,19,31,17,26,32,35,18,12,23,27,21,28,34,36);(1,2,3,4)(7,17,18,21)(12,26,16,28)(19,32,23,34)(27,35,31,36);(5,7)(6,12)(8,21)(9,23)(10,17)(11,18)(13,16)(14,28)(15,31)(19,25)(20,26)(22,32)(24,34)(27,29)(30,35)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55164);