#300361

order	512
nilpotency class	3
isoclinism	isoclinic class of #300333
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	#62477	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

942 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)(3,9)(4,13)(5,7)(6,12)(8,11)(10,15)(14,16)(17,18,23,27)(19,24,30,41)(20,31,33,44)(21,35,22,26)(25,37,43,34)(28,42,45,48)(29,40,36,39)(32,47,46,38);(1,3,4,12,5,10,11,16)(2,6,8,9,7,14,13,15)(17,19)(18,24)(20,32)(21,29)(22,36)(23,30)(25,42)(26,39)(27,41)(28,34)(31,47)(33,46)(35,40)(37,45)(38,44)(43,48);(2,7)(3,10)(8,13)(12,16)(17,20,21,34,23,33,22,37)(18,25,26,31,27,43,35,44)(19,28,29,46,30,45,36,32)(24,38,39,42,41,47,40,48);(1,3,11,16,5,10,4,12)(2,6,13,15,7,14,8,9)(20,33)(25,43)(28,45)(31,44)(32,46)(34,37)(38,47)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=eh,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=dhi,g^c=gi

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, 207557);