#123525

order	512
nilpotency class	3
isoclinism	isoclinic class of #8325
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
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 ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2518 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)(7,16)(8,20,22,13)(9,24)(10,14)(11,15)(12,18)(17,31,33,30)(19,28)(21,29,23,35)(25,27)(26,32,34,36);(1,2)(3,4)(5,7,10,18,11,19,9,25)(6,12,14,28,15,27,24,16)(8,17,23,32,22,33,21,36)(13,26,29,31,20,34,35,30);(2,4)(5,8)(6,13)(7,17)(9,21)(10,23)(11,22)(12,26)(14,29)(15,20)(16,30)(18,32)(19,33)(24,35)(25,36)(27,34)(28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=di,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, 30721);