#118048

order	512
nilpotency class	3
isoclinism	isoclinic class of #10038
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

550 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:

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

Polycyclic presentation:

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

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, 25244);