#272185

order	512
nilpotency class	3
isoclinism	isoclinic class of #272155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3475	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

809 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)(3,12)(4,8)(5,9)(6,11)(7,13)(10,15)(14,16)(17,18,21,26,23,28,22,27)(19,24,30,39,31,40,37,44)(20,32,35,43,36,42,34,25)(29,41,33,47,46,48,45,38);(1,3,6,4)(2,7,11,16)(5,14,15,13)(8,9,12,10)(17,19)(18,24)(20,33)(21,30)(22,37)(23,31)(25,41)(26,39)(27,44)(28,40)(29,34)(32,47)(35,46)(36,45)(38,42)(43,48);(1,4,6,3)(2,8,11,12)(5,14,15,13)(7,9,16,10)(17,20,23,36)(18,25,28,43)(19,29,31,46)(21,34,22,35)(24,38,40,47)(26,42,27,32)(30,45,37,33)(39,48,44,41);(2,9)(5,15)(7,12)(8,16)(10,11)(13,14)(17,21,23,22)(18,26,28,27)(19,30,31,37)(20,34,36,35)(24,39,40,44)(25,42,43,32)(29,45,46,33)(38,48,47,41)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179381);