#205161

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,d^2=h,e^2=hi,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=gi,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112357);