#269194

order	512
nilpotency class	3
isoclinism	isoclinic class of #269147
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60919	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

821 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:

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

Polycyclic presentation:

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

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, 176390);