#485014

order	512
nilpotency class	4
isoclinism	isoclinic class of #484199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#12104	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

730 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392210);