#188169

order	512
nilpotency class	3
isoclinism	isoclinic class of #188062
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2210 conjugacy classes of subgroups; only the 310 normal subgroups are shown.

normal subgroups only

This group has 2210 conjugacy classes of subgroups — too many to draw. Only its 310 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) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=gh,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95365);