#117779

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	C₂×#121	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

531 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,g^2=hi,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, 24975);