#390496

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1065 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=egh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 297692);