#247720

order	512
nilpotency class	3
isoclinism	isoclinic class of #247687
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154916);