#385925

order	512
nilpotency class	3
isoclinism	isoclinic class of #30437
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1379 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293121);