#272585

order	512
nilpotency class	3
isoclinism	isoclinic class of #272539
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	#3481	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

797 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)(3,6)(4,11)(5,7)(8,13)(9,15)(10,14)(12,16)(17,18,24,28,23,30,22,29)(19,31,33,25,21,40,34,27)(20,35,37,26,39,44,38,43)(32,41,47,45,36,42,46,48);(2,7)(3,9)(4,12)(5,15)(8,16)(10,11)(17,19)(18,25)(20,32)(21,23)(22,34)(24,33)(26,41)(27,30)(28,40)(29,31)(35,48)(36,39)(37,47)(38,46)(42,43)(44,45);(1,3,6,2)(4,10,12,8)(5,7,15,9)(11,13,16,14)(17,20)(18,26)(19,32)(21,36)(22,37)(23,39)(24,38)(25,41)(27,42)(28,35)(29,44)(30,43)(31,45)(33,46)(34,47)(40,48);(1,4)(2,8)(3,10)(5,14)(6,12)(7,11)(9,16)(13,15)(17,21,23,19)(18,27,30,25)(20,36,39,32)(22,33,24,34)(26,42,43,41)(28,31,29,40)(35,45,44,48)(37,46,38,47)

Polycyclic presentation:

2^9|a^2=fh,c^2=hi,d^2=i,e^2=h,f^2=i,g^2=hi,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=ghi,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, 179781);