#117778

order	512
nilpotency class	3
isoclinism	isoclinic class of #117771
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,d^c=di,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24974);