#116158

order	512
nilpotency class	3
isoclinism	isoclinic class of #116153
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

562 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)(3,9,11,6)(4,8)(5,7)(10,12,16,13)(14,15)(17,18,21,25,22,26,33,40)(19,23,29,36,28,35,42,46)(20,24,32,39,31,38,44,48)(27,37,41,47,30,34,43,45);(1,2)(3,10,11,16)(4,8)(5,7)(6,12,9,13)(14,15)(17,19)(18,23)(20,30,31,27)(21,29)(22,28)(24,37,38,34)(25,36)(26,35)(32,43,44,41)(33,42)(39,47,48,45)(40,46);(1,3,5,11)(2,6,7,9)(4,13,14,12)(8,16,15,10)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,39)(26,38)(28,30)(29,41)(33,44)(35,37)(36,45)(40,48)(42,43)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,f^b=fhi,d^c=dh,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, 23354);