#285519

order	512
nilpotency class	3
isoclinism	isoclinic class of #285499
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	#18582	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

640 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:

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

Polycyclic presentation:

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

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, 192715);