#285202

order	512
nilpotency class	3
isoclinism	isoclinic class of #285115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

630 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=h,d^2=hi,e^2=h,f^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,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, 192398);