#183717

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₄⋊C₄
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

654 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=i,g^2=h,b^a=be,d^a=dhi,d^b=di,e^b=eh,f^b=fh,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90913);