#230413

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#2361
prime radical	#19125	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

588 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137609);