#300314

order	512
nilpotency class	3
isoclinism	isoclinic class of #300269
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3508	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

913 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207510);