#373701

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2176 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280897);