#230390

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	C₂×#41	#41
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

508 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=de,e^b=eh,g^b=gh,d^c=dh,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, 137586);