#230435

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,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, 137631);