#354312

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

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	C₄×#120
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1790 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261508);