#263028

order	512
nilpotency class	3
isoclinism	isoclinic class of #251809
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	#18126	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

817 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,9)(3,11,13,7)(4,10,6,8)(12,16,14,15)(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);(1,3,5,13)(2,7,9,11)(4,14,6,12)(8,15,10,16)(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);(2,8)(4,6)(7,15)(9,10)(11,16)(12,14)(23,36)(27,39)(37,41)(40,47);(1,4,5,6)(2,8,9,10)(3,12,13,14)(7,16,11,15)(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,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,d^c=di,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170224);