#215023

order	512
nilpotency class	3
isoclinism	isoclinic class of #214768
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	C₂³×C₄	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

821 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,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, 122219);