#230432

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	C₂×#41	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

532 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:

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

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=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, 137628);