#354317

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
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	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1830 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,g^2=i,b^a=be,c^a=cf,d^a=di,f^b=fi,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, 261513);