#328076

order	512
nilpotency class	3
isoclinism	isoclinic class of #327117
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

667 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=i,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235272);