#177706

order	512
nilpotency class	3
isoclinism	isoclinic class of #177385
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

640 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)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27)(33,34)(35,43)(36,39)(37,42)(38,41)(40,46)(44,47)(45,48);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29)(34,39)(36,37)(40,47)(41,42)(43,48)(45,46);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(20,21)(24,29)(26,31)(27,30)(33,35)(34,40)(36,46)(37,45)(38,44)(39,47)(41,48)(42,43);(2,7)(6,15)(8,12)(9,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)(34,41)(35,44)(39,42)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 84902);