#230421

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

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=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, 137617);