#230433

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
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	#2361
prime radical	#3035	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

561 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137629);