#263022

order	512
nilpotency class	3
isoclinism	isoclinic class of #252247
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2408
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

772 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170218);