#215084

order	512
nilpotency class	3
isoclinism	isoclinic class of #214736
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	#36032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

878 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:

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

Polycyclic presentation:

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

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, 122280);