#181836

order	512
nilpotency class	2
isoclinism	isoclinic class of #180872
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#376
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

608 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 314 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)(19,27,30,40)(20,24)(21,25)(22,26)(23,29,35,43)(28,39,41,47)(31,36)(32,37)(33,38)(34,42,45,48)(44,46);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(17,19)(18,23,26,35)(20,28)(21,29)(22,30)(24,34,36,45)(25,27,38,40)(31,41)(32,42)(33,43)(37,39,46,47)(44,48);(3,10)(7,14)(11,12)(15,16)(17,20,21,32,22,31,33,44)(18,24,25,37,26,36,38,46)(19,28,29,42,30,41,43,48)(23,34,27,39,35,45,40,47);(1,4,6,5)(2,8,9,13)(3,11,10,12)(7,15,14,16)(20,31)(24,36)(28,41)(32,44)(34,45)(37,46)(39,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89032);