#316188

order	512
nilpotency class	3
isoclinism	isoclinic class of #316187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

428 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=e,c^2=ef,d^2=e,e^2=h,f^2=hi,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,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, 223384);