#211753

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×D₄	#41
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118949);