#317214

order	512
nilpotency class	3
isoclinism	isoclinic class of #317195
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

509 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)(4,8)(5,7)(6,11)(9,13)(12,14)(15,16)(17,18,23,26)(19,27,30,45)(20,25,35,43)(21,22,36,37)(24,38,40,29)(28,41,47,44)(31,39,46,48)(32,33,42,34);(1,3)(2,6)(4,11)(5,12)(7,15)(8,10)(9,14)(13,16)(17,19,22,29,23,30,37,38)(18,24,21,27,26,40,36,45)(20,31,34,47,35,46,33,28)(25,41,42,48,43,44,32,39);(2,7)(3,11)(5,13)(6,14)(8,9)(10,15)(17,20)(18,25)(19,28)(21,32)(22,33)(23,35)(24,39)(26,43)(27,44)(29,46)(30,47)(31,38)(34,37)(36,42)(40,48)(41,45);(2,8)(3,11)(7,9)(12,16)(17,21,22,26,23,36,37,18)(19,27,29,40,30,45,38,24)(20,32,33,43,35,42,34,25)(28,44,46,48,47,41,31,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,f^b=fh,f^c=fi,g^c=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, 224410);