#213816

order	512
nilpotency class	3
isoclinism	isoclinic class of #213808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

856 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121012);