#482182

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

893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18,21,26,22,24,34,39)(19,27,29,42,30,23,35,37)(20,25,32,41,33,38,45,47)(28,40,43,48,31,36,44,46);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,30)(18,23,24,27)(20,31,33,28)(21,29,34,35)(25,40,38,36)(26,42,39,37)(32,44,45,43)(41,46,47,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,24)(21,34)(23,27)(25,38)(29,35)(32,45)(36,40)(43,44);(17,20,22,33)(18,25,24,38)(19,28,30,31)(21,32,34,45)(23,36,27,40)(26,41,39,47)(29,43,35,44)(37,46,42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cgh,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, 389378);