#316191

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

Polycyclic presentation:

2^9|a^2=egi,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, 223387);