#208581

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=fh,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=dhi

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, 115777);