#227601

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#18494	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 134797);