#245877

order	512
nilpotency class	3
isoclinism	isoclinic class of #245807
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#45388	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,e^c=ei,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153073);