#214807

order	512
nilpotency class	3
isoclinism	isoclinic class of #214704
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#392	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122003);