#123570

order	512
nilpotency class	3
isoclinism	isoclinic class of #105681
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2478 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:

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

Polycyclic presentation:

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

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, 30766);