#248823

order	512
nilpotency class	3
isoclinism	isoclinic class of #248743
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	C₂³×C₄	C₂⁴
prime radical	#3505	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1029 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156019);