#353446

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3990 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316921, #72316933, #72316946
maximal normal subgroup of	#72316921, #72316933, #72316946

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260642);