#230417

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	#2361
prime radical	#3035	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

555 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:

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

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,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, 137613);