#214652

order	512
nilpotency class	3
isoclinism	isoclinic class of #214040
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

835 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)(3,10,12,6)(4,9)(5,8)(7,11,16,13)(14,15)(17,18)(19,23)(20,30,32,24)(21,25)(22,26)(27,41,43,35)(28,36)(29,37)(31,44,46,38)(33,39)(34,40)(42,48,45,47);(1,3)(2,6)(4,13)(5,12)(7,15)(8,10)(9,16)(11,14)(17,19,22,29)(18,23,26,37)(20,27,32,43)(21,33,34,28)(24,35,30,41)(25,39,40,36)(31,45,46,42)(38,48,44,47);(1,3)(2,7)(4,13)(5,12)(6,9)(8,16)(10,15)(11,14)(17,20)(18,24)(19,27)(21,31)(22,32)(23,35)(25,38)(26,30)(28,42)(29,43)(33,45)(34,46)(36,47)(37,41)(39,48)(40,44);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,14)(15,16)(17,21,22,34)(18,25,26,40)(19,28,29,33)(20,31,32,46)(23,36,37,39)(24,38,30,44)(27,42,43,45)(35,47,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=h,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=fh,g^c=ghi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121848);