#285111

order	512
nilpotency class	3
isoclinism	isoclinic class of #285051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

625 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:

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

Polycyclic presentation:

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

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, 192307);