#152837

order	512
nilpotency class	5
isoclinism	isoclinic class of #13700
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#508
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

208 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=d,d^2=gh,e^2=gi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gh,h^a=hi,e^c=ei

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, 60033);