#488609

order	512
nilpotency class	4
isoclinism	isoclinic class of #488523
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
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

766 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)(19,28)(20,25)(21,33)(22,37)(23,26)(24,30)(27,35)(29,44)(31,45)(32,42)(34,39)(36,47)(38,43)(40,46)(41,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19)(18,24)(20,29)(21,34)(22,38)(23,31)(25,39)(26,40)(27,43)(28,33)(30,35)(32,46)(36,48)(37,41)(42,44)(45,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20,23,22)(18,25,26,37)(19,29,31,38)(21,32,36,35)(24,39,40,41)(27,33,42,47)(28,44,45,43)(30,34,46,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(18,26)(24,40)(25,37)(27,42)(28,45)(33,47)(39,41)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395805);