#488605

order	512
nilpotency class	4
isoclinism	isoclinic class of #488515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#12102	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

774 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:

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

Polycyclic presentation:

2^9|d^2=hi,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395801);