#230408

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
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

559 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)(3,11,12,8,15,16,14,13)(17,18,24,28)(19,29)(20,21,35,38)(22,27,23,40)(25,39)(26,41,37,34)(30,45)(31,47)(32,44)(33,43)(36,42)(46,48);(1,3)(2,8)(4,12)(5,16)(6,14)(7,15)(9,11)(10,13)(17,19,23,32,24,33,22,39)(18,25,40,29,28,44,27,43)(20,30,41,46,35,47,34,42)(21,36,37,45,38,48,26,31);(1,4,7,6)(2,5,10,9)(3,12,15,14)(8,16,13,11)(17,20,24,35)(18,26,28,37)(19,30,33,47)(21,27,38,40)(22,34,23,41)(25,31,44,45)(29,36,43,48)(32,46,39,42);(1,5,4,10,7,9,6,2)(3,13,14,16,15,8,12,11)(17,21,22,26,24,38,23,37)(18,20,27,34,28,35,40,41)(19,31,32,36,33,45,39,48)(25,42,29,30,44,46,43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=e,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, 137604);