#263553

order	512
nilpotency class	3
isoclinism	isoclinic class of #263527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62342	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(18,23)(20,33)(21,36)(24,38)(25,41)(26,27)(28,40)(29,45)(30,31)(34,46)(35,47)(44,48);(2,6)(5,12)(9,16)(11,14)(17,18,21,23,22,26,36,27)(19,28,30,40,32,38,31,24)(20,25,33,37,35,41,47,43)(29,44,45,42,34,48,46,39);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,19)(18,24)(20,34)(21,30)(22,32)(23,28)(25,42)(26,40)(27,38)(29,35)(31,36)(33,46)(37,48)(39,41)(43,44)(45,47);(2,6)(7,15)(8,13)(9,16)(17,20,21,33,22,35,36,47)(18,25,23,37,26,41,27,43)(19,29,30,45,32,34,31,46)(24,39,28,44,40,42,38,48)

Polycyclic presentation:

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

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, 170749);