#152205

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	#2163
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

308 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6,9,16,10,17,28,40,11,15,29,38,30,39,54,60,12,14,27,36,31,37,53,59,8,24,25,50,26,51,52,66)(7,18,21,42,22,43,47,63,23,33,48,56,49,57,65,67,19,41,44,61,45,62,64,68,20,13,32,34,46,35,55,58);(1,2,3,4)(5,7,8,20,12,19,11,23)(6,13,15,18,14,33,24,41)(9,21,25,32,27,44,29,48)(10,22,26,46,31,45,30,49)(16,34,38,42,36,56,50,61)(17,35,39,43,37,57,51,62)(28,47,52,55,53,64,54,65)(40,58,60,63,59,67,66,68);(6,14)(7,19)(9,27)(15,24)(17,37)(20,23)(22,45)(25,29)(28,53)(34,56)(39,51)(42,61)(46,49)(52,54)(58,67)(63,68)

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=ci,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59401);