#147370

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

714 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54566);