#191301

order	512
nilpotency class	3
isoclinism	isoclinic class of #188768
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#383
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

488 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=e,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,d^b=dh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98497);