#230420

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#19125	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

584 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)(3,11,12,13,14,15,16,8)(17,18)(19,30,34,44)(20,35)(21,27)(22,28)(23,36)(24,29)(25,33,45,40)(26,37)(31,32,46,42)(38,48,47,43)(39,41);(1,3)(2,8)(4,12)(5,15)(6,16)(7,14)(9,11)(10,13)(17,19,23,33,24,34,22,40)(18,25,36,30,29,45,28,44)(20,31,41,43,27,46,37,48)(21,38,26,42,35,47,39,32);(1,4,7,6)(2,9,10,5)(3,12,14,16)(8,11,13,15)(17,20,24,27)(18,26,29,39)(19,31,34,46)(21,36,35,28)(22,37,23,41)(25,42,45,32)(30,47,44,38)(33,43,40,48);(1,5,6,10,7,9,4,2)(3,13,12,15,14,8,16,11)(17,21,22,39,24,35,23,26)(18,27,28,41,29,20,36,37)(19,32,33,38,34,42,40,47)(25,43,30,46,45,48,44,31)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=e,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, 137616);