#482762

order	512
nilpotency class	4
isoclinism	isoclinic class of #482734
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	#41419	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#553 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

832 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)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,21,26,22,25,34,41)(19,27,30,42,29,23,35,37)(20,24,33,40,32,39,45,48)(28,38,43,47,31,36,44,46);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,22,29)(18,23,25,27)(20,31,32,28)(21,30,34,35)(24,38,39,36)(26,42,41,37)(33,44,45,43)(40,46,48,47);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,22,32)(18,24,25,39)(19,28,29,31)(21,33,34,45)(23,36,27,38)(26,40,41,48)(30,43,35,44)(37,46,42,47);(3,10)(7,14)(11,16)(13,15)(19,29)(20,32)(23,27)(24,39)(30,35)(33,45)(37,42)(40,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgh,d^c=dhi,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, 389958);