#356843

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(6,10)(11,12)(13,14,18,22,19,23,17,32)(15,24,26,20,27,35,33,36)(16,21,31,39,28,37,30,44)(25,40,41,34,29,42,43,38);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(16,28)(21,37)(25,29)(30,31)(34,38)(39,44)(40,42)(41,43);(5,7,9,8)(6,11,10,12)(13,15,18,26,19,27,17,33)(14,20,22,35,23,36,32,24)(16,29,31,43,28,25,30,41)(21,38,39,40,37,34,44,42);(5,8,9,7)(6,11,10,12)(13,16,18,31,19,28,17,30)(14,21,22,39,23,37,32,44)(15,25,26,41,27,29,33,43)(20,34,35,42,36,38,24,40)

Polycyclic presentation:

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

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, 264039);