#272346

order	512
nilpotency class	3
isoclinism	isoclinic class of #272283
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	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

832 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)(3,13)(4,11)(5,8)(6,9)(7,12)(10,14)(15,16)(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);(1,3,7,14)(2,8,12,16)(4,5,6,15)(9,13,11,10)(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);(1,4,7,6)(2,9,12,11)(3,5,14,15)(8,13,16,10)(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);(1,5,7,15)(2,10,12,13)(3,4,14,6)(8,11,16,9)(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=fh,b^2=h,c^2=hi,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179542);