#358285

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 265481);