#317511

order	512
nilpotency class	3
isoclinism	isoclinic class of #317499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

547 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224707);