#116117

order	512
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

581 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)(4,12)(5,8)(6,10)(7,13)(11,16)(14,15)(17,18,22,28,23,27,21,26)(19,24,32,40,31,41,30,39)(20,33,37,25,35,43,36,42)(29,44,46,38,34,47,45,48);(4,13)(6,15)(7,12)(10,14)(17,19)(18,24)(20,34,35,29)(21,30)(22,32)(23,31)(25,38,42,48)(26,39)(27,41)(28,40)(33,44,43,47)(36,46,37,45);(1,3,5,11)(2,6,8,15)(4,14,13,10)(7,16,12,9)(17,20)(18,25)(19,29)(21,37)(22,36)(23,35)(24,38)(26,43)(27,42)(28,33)(30,46)(31,34)(32,45)(39,47)(40,44)(41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,e^2=hi,g^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=ehi,g^c=gh

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, 23313);