#164115

order	512
nilpotency class	3
isoclinism	isoclinic class of #163983
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#2615	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

970 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)(3,7,8,5)(4,6)(9,10)(11,15)(12,14)(13,17)(16,20)(18,22)(19,21)(23,24)(25,26,29,33,30,34,28,32)(27,31,37,40,35,38,36,39);(1,3,4,8)(2,5,6,7)(10,14)(12,20)(15,21)(16,17)(19,24)(22,23)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,40)(34,38);(9,11,13,18)(10,15,17,22)(12,19,20,24)(14,21,16,23)(27,35)(31,38)(36,37)(39,40);(11,18)(15,22)(19,24)(21,23)(25,28,30,29)(26,32,34,33)(27,36,35,37)(31,39,38,40)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=h,d^2=i,e^2=h,f^2=i,b^a=be,e^a=eg,c^b=ci,e^b=eh,f^b=fgh,d^c=dh

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, 71311);