#482529

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×C₂²⋊C₄	C₂⁴
prime radical	#36032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

871 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)(19,27)(20,24)(21,35,36,25)(22,26)(23,30,39,29)(28,43)(31,38)(32,41,33,40)(34,42)(37,44,47,46)(45,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,31)(18,23,26,39)(20,28,34,45)(21,30,36,29)(24,37,42,47)(25,27,35,38)(32,46,33,44)(40,48,41,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20)(18,24)(19,28)(21,33)(22,34)(23,37)(25,41)(26,42)(27,43)(29,46)(30,44)(31,45)(32,36)(35,40)(38,48)(39,47);(17,21)(18,25)(19,29)(20,32)(22,36)(23,38)(24,40)(26,35)(27,39)(28,44)(30,31)(33,34)(37,43)(41,42)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,d^b=di,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	#72318754
maximal normal subgroup of	#72318754

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389725);