#214058

order	512
nilpotency class	3
isoclinism	isoclinic class of #214040
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 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:

(1,2,7,10,6,11,5,14)(3,9,12,15,4,8,13,16)(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)(2,8)(4,6)(5,13)(7,12)(9,11)(10,16)(14,15)(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);(1,4,6,3)(2,9,11,8)(5,13,7,12)(10,16,14,15)(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);(1,5,6,7)(2,10,11,14)(3,12,4,13)(8,15,9,16)(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,c^2=i,d^2=hi,f^2=h,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=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, 121254);