#482536

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	#392	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

829 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)(17,18,21,26,22,23,35,37)(19,24,28,39,29,36,45,46)(20,30,32,42,33,25,34,41)(27,40,43,48,31,38,44,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(18,23)(24,36)(25,30)(38,40);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,19,22,29)(18,24,23,36)(20,31,33,27)(21,28,35,45)(25,40,30,38)(26,39,37,46)(32,44,34,43)(41,48,42,47);(17,20,22,33)(18,25,23,30)(19,27,29,31)(21,34,35,32)(24,38,36,40)(26,42,37,41)(28,44,45,43)(39,48,46,47)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=i,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,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.

Relationships

maximal subgroup of	#72318712
maximal normal subgroup of	#72318712

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389732);