#481804

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#36032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

929 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,d^c=di,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389000);