#481706

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1066 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)(9,10,14,19,15,18,28,34)(11,20,22,35,23,16,29,32)(12,24,25,36,13,17,27,33)(21,30,37,39,26,31,38,40);(1,2)(3,6)(4,8)(5,7)(9,11,15,23)(10,16,18,20)(12,21,13,26)(14,22,28,29)(17,30,24,31)(19,35,34,32)(25,37,27,38)(33,40,36,39);(1,3,5,4)(2,6,7,8)(9,12,15,13)(10,17,18,24)(11,21,23,26)(14,27,28,25)(16,30,20,31)(19,36,34,33)(22,38,29,37)(32,40,35,39);(9,13,15,12)(10,17,18,24)(11,21,23,26)(14,27,28,25)(16,31,20,30)(19,33,34,36)(22,37,29,38)(32,40,35,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,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, 388902);