#117955

order	512
nilpotency class	3
isoclinism	isoclinic class of #117951
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	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25151);