#386794

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1220 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fgi,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi

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, 293990);