#483425

order	512
nilpotency class	4
isoclinism	isoclinic class of #483425
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216671, #72216672, #72216898, …
maximal normal subgroup of	#72216671, #72216672, #72216898, …
maximal quotient of	C₂×#483425, #72318563, #72318811, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390621);