#180697

order	512
nilpotency class	3
isoclinism	isoclinic class of #180581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#53347	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,d^b=dhi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87893);