#482707

order	512
nilpotency class	4
isoclinism	isoclinic class of #482397
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

898 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)(3,4)(7,8)(9,10,13,18,14,17,26,33)(11,19,21,34,22,15,27,29)(12,16,25,32,24,31,37,40)(20,30,35,39,23,28,36,38);(1,2)(3,6)(4,8)(5,7)(9,11,14,22)(10,15,17,19)(12,23,24,20)(13,21,26,27)(16,30,31,28)(18,34,33,29)(25,36,37,35)(32,38,40,39);(1,3,5,4)(2,6,7,8)(9,12,14,24)(10,16,17,31)(11,20,22,23)(13,25,26,37)(15,28,19,30)(18,32,33,40)(21,35,27,36)(29,38,34,39);(1,3,5,4)(2,6,7,8)(10,17)(12,24)(13,26)(15,19)(20,23)(21,27)(32,40)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389903);