#180700

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

611 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=gh,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, 87896);