#356751

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1316 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 263947);