#354475

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1494 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)(7,17,21,32)(8,13,23,30)(9,14)(10,15)(11,16)(12,20,29,26)(18,25)(19,27,33,22)(24,31)(28,34)(35,36);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,26)(11,21)(13,24)(14,23)(15,32)(16,29)(17,34)(20,28)(22,31)(25,33)(27,35)(30,36);(5,8)(6,13)(7,18)(9,17)(10,27)(11,23)(12,24)(14,21)(15,33)(16,30)(19,34)(20,31)(22,28)(25,32)(26,35)(29,36);(5,9,10,24,11,25,28,36)(6,14,15,31,16,18,34,35)(7,19,26,13,21,33,20,30)(8,17,27,12,23,32,22,29)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261671);