#208803

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#537	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

818 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115999);